# Galaxy bias renormalization group

Henrique Rubira  Fabian Schmidt

###### Abstract

The effective field theory of large-scale structure allows for a consistent
perturbative bias expansion of the rest-frame galaxy density field. In this
work, we present a systematic approach to renormalize galaxy bias and
stochastic parameters using a finite cutoff scale $\Lambda$. We derive the
differential equations of the Wilson-Polchinski renormalization group that
describe the evolution of the finite-scale bias parameters with $\Lambda$,
analogous to the $\beta$-function running in QFT. We further provide the
connection between the finite-cutoff scheme and the renormalization procedure
for $n$-point functions that has been used as standard in the literature so
far; some inconsistencies in the treatment of renormalized bias in current EFT
analyses are pointed out as well. The fixed-cutoff scheme allows us to
predict, in a principled way, the finite part of loop contributions which is
due to perturbative modes and which, in the standard renormalization approach,
is absorbed into counterterms. We expect that this will allow for the robust
extraction of (a yet-to-be-determined amount of) additional cosmological
information from galaxy clustering, both when using field-level techniques and
$n$-point functions.

##  1 Introduction

The large-scale structure of the universe (LSS), as probed via the three-
dimensional distribution of galaxies, is known to be a rich source of
information on gravity, dark matter, dark energy, and the initial conditions
of structure formation. To unlock this information, however, one has to
marginalize over the significant uncertainties in our understanding of galaxy
formation. What is required in particular is a reliable prediction for the
conditional probability of forming a galaxy that passes observational
luminosity and color selections, at a given location. The effective field
theory (EFT) approach [1, 2, 3, 4], built upon cosmological perturbation
theory, allows for a consistent expansion of the galaxy density field into
operators $O$, ranked in terms of perturbative order, and corresponding bias
coefficients $b_{O}$ as well as stochastic contributions described by a field
$\epsilon$ and coefficients $c_{\epsilon,O}$ (see [5] for a review):

$\delta_{g}(\bm{x},\tau)\equiv\frac{n_{g}(\bm{x},\tau)}{\bar{n}_{g}(\tau)}-1=\sum_{O}\left[b_{O}(\tau)+c_{\epsilon,O}(\tau)\epsilon(\bm{x},\tau)\right]O(\bm{x},\tau)+\epsilon(\bm{x},\tau).$
(1.1)

The bias parameters and stochastic coefficients are only functions of time
$\tau$, while the stochastic field does not correlate with the operators $O$
and is characterized completely by its moments [6]. The bias operators are
constructed out of the matter density field $\delta(\bm{x},\tau)=\rho_{\rm
m}(\bm{x},\tau)/\bar{\rho}_{\rm m}(\tau)-1$. It is a nontrivial result that a
local-in-time relation such as Eq. (1.1) can also capture the long time scales
over which galaxies form [6, 7] (see also [8]), i.e. despite the fact the
theory is fundamentally nonlocal in time. This is a consequence of the fact
that, at any given order in perturbation theory, the solutions to the equation
of motion for matter and gravitational potential are a sum over a finite
number of terms which each are factorizable functions of time and space [9,
10].

When computing observable statistics such as $n$-point correlation functions
based on Eq. (1.1) (technically these involve expectation values of products
of composite operators), loop integrals arise which have to be regularized.
The standard approach has so far been to work at the level of $n$-point
correlation functions, and to remove the UV-sensitive loop integrals via
counterterms [11]; for recent work on different regularization schemes for the
galaxy bias, see [12]. In this paper, our aim is instead to follow the
renormalization-group (RG) approach of Wilson [13] and Polchinski [14], which
proceeds by integrating out small-scale modes in the partition function.111Two
main different RG approaches are used in the context of field theory: the
Wilsonian RG and the continuum RG [15]. The Wilsonian RG makes use of the fact
that the physics at energy scales $E\ll\Lambda$ has to be independent of the
finite cutoff $\Lambda$ (although the couplings may change with $\Lambda$).
Later, Polchinski derived the Wilsonian equations in terms of differential
increments in the path integral formalism [14], an approach similar to the one
taken in this work. The continuum case instead makes direct use of the
invariance of an observable with respect to any arbitrary scale, e.g. via
dimensional regularization [16, 17, 18, 19]. It is more frequently used in the
field theory context since it usually allows for more straightforward
calculations than the Wilson/Polchinski approach. We will see here however
that the latter can be readily applied in the EFT of LSS context. Ref. [3] was
the first to point out how this approach can be adapted to the EFT of LSS.
Ref. [20] used it to derive the EFT prediction for the conditional probability
density of the galaxy field given the matter density field; that is, the
crucial ingredient needed for making predictions for galaxy clustering, as
noted above.

The partition function is usually formulated in terms of a Lagrangian for the
fields under consideration. However, the nonlinear dynamics of collisionless
matter under gravity cannot be derived from an interaction Lagrangian written
in terms of three-dimensional fields. One can reverse-engineer a Lagrangian
that gives rise to the fluid equations [21, 22]. This Lagrangian will however
not give rise to the corrections to a perfect fluid, such as effective
pressure and stochastic fluctuations, which are expected in the EFT. It is
even less applicable to generate the EFT for biased tracers who do not obey
mass and momentum conservation. Instead, our partition function will be based
directly on the bias expansion Eq. (1.1), and the free propagator for the
linear density field, that is the linear power spectrum $P_{\rm L}(k)$.

The Wilson-Polchinski approach to renormalization proceeds by integrating out
modes in the free field above a momentum cutoff $\Lambda$. In the LSS case,
this corresponds to integrating modes above the cutoff in the linear density
field. This is in fact essential in order to keep predictions for the galaxy
density field under perturbative control, as noted in [23]. In this paper, we
continue to work along these lines by deriving how the bias coefficients
$b_{O}^{\Lambda}$ (and stochastic amplitudes), which now depend explicitly on
the finite scale $\Lambda$, evolve under a change in the cutoff. Following
Wilson’s foundational work on this topic, this is known as “renormalization
group flow” in the field theory context.

Our results can be useful in several ways:

  * •

Observational measurements of bias coefficients $b_{O}^{\Lambda}$ at different
$\Lambda$ can be compared with the EFT prediction for the running, providing
an additional cross-check of EFT-based cosmology inferences based on the same
data.

  * •

Simulation measurements of bias coefficients are typically performed using
$n$-point correlation functions on large scales (e.g., [24]) or the separate-
universe technique [25, 26, 27], both of which return bias coefficients in the
low-scale limit $\Lambda\to 0$. If one wants to use these measurements to put
prior information on finite-scale bias parameters, one needs to run the former
to the desired finite $\Lambda$ (see Sec. 2 and Sec. 3).

  * •

We point out that the finite-scale renormalization scheme allows us to predict
finite contributions to loop integrals that are completely absorbed by
counterterms in the approach used so far (Sec. 4). This indicates that there
might be more cosmological information to be gained when adopting the finite-
scale renormalization instead.

The outline of the paper is as follows. We first introduce our notation and
conventions. In Sec. 2 we derive the RG equations for the galaxy bias and
stochastic parameters, discussing particular solutions for the bias evolution
in $\Lambda$. We connect the finite-$\Lambda$ renormalization scheme to the
$n$-point renormalized bias in Sec. 3 and discuss in Sec. 4 how the two
schemes treat loop contributions. We conclude in Sec. 5. The appendices
contain details on the more technical calculations. We use a Planck 2018
Euclidean $\Lambda$CDM cosmology [28] for all numerical results.

### Dictionary and Notation

We provide here a brief explanation of the terms and notations used in this
paper.

#### Fourier conventions.

We keep the letters $\bm{k},\bm{p}$ and $\bm{q}$ for momenta variables. We use
bold letters for three-vectors and subscripts to refer to sum of vectors, e.g.
$\bm{p}_{1\dots n}=\bm{p}_{1}+\dots+\bm{p}_{n}$. We use the following notation
for the Fourier-space integrals

$\int_{\bm{p}_{1},\dots,\bm{p}_{n}}=\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\dots\int\frac{d^{3}p_{n}}{(2\pi)^{3}}\,.$
(1.2)

The smoothing of a field $f$ on a length scale $1/\Lambda$ with a filter $W$
is done in real space via

$f_{\Lambda}(\bm{x})=\int
d^{3}x^{\prime}\,W_{\Lambda}(\bm{x}^{\prime}-\bm{x})f(\bm{x}^{\prime})\,,$
(1.3)

such that in Fourier space

$f_{\Lambda}(\bm{k})=\,W_{\Lambda}(\bm{k})f(\bm{k})\,.$ (1.4)

Throughout this work we assume the filter $W_{\Lambda}$ to be a sharp filter
in Fourier space.

#### Scales.

  * •

$\Lambda$: denotes the smoothing scale when not fixed to any value;

  * •

$\Lambda_{\ast}$: is the fixed reference value of the smoothing scale, i.e.
the fixed cutoff used in a given analysis;

  * •

$k_{\rm NL}$: is the range of validity of the theory, where perturbation
theory breaks down;

  * •

$k_{\rm max}$: is the highest wavenumber considered in a given analysis.

Within the perturbative RG flow context, the hierarchy $k_{\rm
max}<\Lambda,\Lambda_{\ast}<k_{\rm NL}$ has to be fulfilled.

#### Operators.

  * •

$O[\delta]$: General operator, which is constructed as a convolution of
density fields

$O[\delta](\bm{k})=\int_{\bm{p}_{1},\ldots,\bm{p}_{n}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1\ldots
n})S_{O}(\bm{p}_{1},\ldots\bm{p}_{n})\delta(\bm{p}_{1})\cdots\delta(\bm{p}_{n}).$
(1.5)

We refer to the number of density fields appearing in the definition as
“order” of the operator. Hence in case of Eq. (1.5), the order is $n$, and we
also denote this as $O^{[n]}$.

  * •

We work with the following basis of operators in Eq. (1.1):

First order: $\displaystyle\quad\delta\,;$ Second order:
$\displaystyle\quad\delta^{2},\ \mathcal{G}_{2}\,;$ Third order:
$\displaystyle\quad\delta^{3},\ \delta\,\mathcal{G}_{2},\ \Gamma_{3},\
\mathcal{G}_{3}\,;$ (1.6)

where the Galileon operators are defined in terms of the scaled gravitational
potential $\Phi_{g}\equiv\nabla^{-2}\delta$ via

$\displaystyle\mathcal{G}_{2}(\Phi_{g})$ $\displaystyle\equiv$
$\displaystyle(\nabla_{i}\nabla_{j}\Phi_{g})^{2}-(\nabla^{2}\Phi_{g})^{2}\,,$
(1.7) $\displaystyle\mathcal{G}_{3}(\Phi_{g})$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}\left[2\nabla_{i}\nabla_{j}\Phi_{g}\nabla^{j}\nabla_{k}\Phi_{g}\nabla^{k}\nabla^{i}\Phi_{g}+(\nabla^{2}\Phi_{g})^{3}-3(\nabla_{i}\nabla_{j}\Phi_{g})^{2}\nabla^{2}\Phi_{g}\right]\,.$
(1.8)

The operator

$\displaystyle\Gamma_{3}(\Phi_{g},\Phi_{v})\equiv\mathcal{G}_{2}(\Phi_{g})-\mathcal{G}_{2}(\Phi_{v})$
(1.9)

is constructed out of the gravitational, $\Phi_{g}$, and velocity potential,
$\Phi_{v}\equiv\nabla^{-2}\bm{\nabla}\cdot\bm{v}$.

We use the notation
$\sigma^{2}_{\bm{k}_{1},\bm{k}_{2}}=\left(\bm{k}_{1}\cdot\bm{k}_{2}/k_{1}k_{2}\right)^{2}-1$
to define $\mathcal{G}_{2}$ in Fourier space, i.e. [cf. Eq. (1.5) above]

$S_{\mathcal{G}_{2}}(\bm{k}_{1},\bm{k}_{2})=\sigma^{2}_{\bm{k}_{1},\bm{k}_{2}}.$
(1.10)

  * •

The evolved matter density field $\delta$ is expanded in terms of the linear
field $\delta^{(1)}$ via

$\delta(\bm{k})=\sum_{\ell=1}^{\infty}\int_{\bm{p}_{1},\ldots,\bm{p}_{\ell}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1\ldots\ell})F_{\ell}(\bm{p}_{1},\ldots\bm{p}_{\ell})\delta^{(1)}(\bm{p}_{1})\cdots\delta^{(1)}(\bm{p}_{\ell}),$
(1.11)

with $F_{\ell}$ being the usual PT kernels [29], including $F_{1}\equiv 1$.

  * •

We denote the contribution to a given operator $O$ at $m$-th order as
$O^{(m)}$, so if $O=O^{[n]}$, then $m\geq n$. In other words, superscripts in
parenthesis refer to the expansion order of an operator and superscripts in
square brackets refer to the order of an operator.

  * •

$O[\delta_{\Lambda}^{(1)}]$: same operator as Eq. (1.5) above, but constructed
by smoothing the initial conditions at a scale $\Lambda$. Note that we use the
shorthand notation $O[\delta_{\Lambda}^{(1)}]\equiv
O[\delta[\delta_{\Lambda}^{(1)}]]$ for operators constructed from the forward-
evolved matter density field, which in turn is initialized with the cut linear
density field. This should not be understood as operators directly constructed
from the linear density field.

  * •

$\llbracket O\rrbracket$: the $n$-point function renormalized operator $O$
introduced by [11] and defined using the renormalization conditions, for all
$m\geq 0$,

$\langle\delta^{(1)}(\bm{k}_{1})\cdots\delta^{(1)}(\bm{k}_{m})\llbracket
O\rrbracket(\bm{k})\rangle\stackrel{{\scriptstyle k_{i}\to
0}}{{\longrightarrow}}\langle\delta^{(1)}(\bm{k}_{1})\cdots\delta^{(1)}(\bm{k}_{m})O[\delta](\bm{k})\rangle_{\rm
LO}\,,$ (1.12)

where LO indicates the leading order term (see Sec. 3). The renormalized
$\delta^{2}$ operator, $\llbracket\delta^{2}\rrbracket$, at 1-loop and up to
$m=2$ reads

$\llbracket\delta^{2}\rrbracket=\delta^{2}-\sigma^{2}_{\infty}\left(1+\frac{68}{21}\delta+\frac{8126}{2205}\delta^{2}+\frac{254}{2205}\mathcal{G}_{2}\right)\,,$
(1.13)

where $\sigma^{2}_{\infty}=\lim_{\Lambda\to\infty}\sigma^{2}_{\Lambda}$ [see
Eq. (1.21)]. For a broader discussion on different renormalization schemes,
see [12].

  * •

We expand the power spectrum of the stochastic field $\epsilon$ in Eq. (1.1)
as (cf. [5])

$\langle\epsilon(\bm{k})\epsilon(\bm{k}^{\prime})\rangle=(2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})\sum_{n=0}^{\infty}P_{\epsilon}^{\\{2n\\}}k^{2n},$
(1.14)

where we also often abbreviate $P_{\epsilon}\equiv P_{\epsilon}^{\\{0\\}}$.
Note that $\langle\epsilon(\bm{k})\epsilon(\bm{k}^{\prime})\rangle$
corresponds also to the stochastic contribution to the galaxy power spectrum.

#### Partition function.

  * •

$\mathcal{P}[\varphi]$: The probability distribution function (PDF) of a field
$\varphi$, which enters as a measure of the path integral. If $\varphi$ is
assumed to be Gaussian, then

$\mathcal{P}[\varphi]=\left(\prod_{\bm{k}}^{\Lambda}2\pi
P_{\varphi}(k)\right)^{-1/2}\exp\left[-\frac{1}{2}\int_{\bm{k}}^{\Lambda}\frac{|\varphi|^{2}}{P_{\varphi}(k)}\right]\,,$
(1.15)

where $P_{\varphi}$ is the power spectrum of $\varphi$.

  * •

$\mathcal{Z}[J_{\Lambda}]$ : The partition function evaluated on a current
with support over wavenumbers up to $\Lambda$. We have that

$\mathcal{Z}[J_{\Lambda}]=\int\mathcal{D}\delta^{(1)}_{\Lambda}\,\mathcal{P}[\delta^{(1)}_{\Lambda}]\exp\left(S_{\rm
int}[\delta^{(1)}_{\Lambda},J_{\Lambda}]\right)\,,$ (1.16)

where $S_{\rm int}[\delta^{(1)}_{\Lambda},J_{\Lambda}]$ is the interaction
action.

#### Initial conditions.

We assume Gaussian initial conditions throughout [cf. Eq. (1.15)]. We keep
only the fastest growing mode, allowing us to phrase the initial conditions
completely in terms of $\varphi=\delta^{(1)}$. We consider three types of
power spectra (or propagators) for $\delta^{(1)}$: the linear propagator

$P_{\rm
L}(k)=\langle\delta^{(1)}(\bm{k})\delta^{(1)}(\bm{k}^{\prime})\rangle^{{}^{\prime}},$
(1.17)

the linear propagator cut at $\Lambda$

$P_{\rm
L}^{\Lambda}(k)=\langle\delta^{(1)}_{\Lambda}(\bm{k})\delta^{(1)}_{\Lambda}(\bm{k}^{\prime})\rangle^{{}^{\prime}}=\langle\delta^{(1)}_{\Lambda}(\bm{k})\delta^{(1)}(\bm{k}^{\prime})\rangle^{{}^{\prime}},$
(1.18)

where we used the sharp-$k$ cutoff properties in the second equality, and the
shell propagator

$P_{\rm shell}(k)=\langle\delta^{(1)}_{\rm shell}(\bm{k})\delta^{(1)}_{\rm
shell}(\bm{k}^{\prime})\rangle^{{}^{\prime}}$ (1.19)

which only has support within an infinitesimal shell
$[\Lambda,\Lambda+\lambda]$ in momentum space. Throughout, we use the primed
notation,

$\langle O(\bm{k}_{1})\dots O(\bm{k}_{n})\rangle=(2\pi)^{3}\,\delta_{\rm
D}(\bm{k}_{1\dots n})\,\langle O(\bm{k}_{1})\dots
O(\bm{k}_{n})\rangle^{\prime}$ (1.20)

for ensemble averages with the overall momentum constraint removed.

We also define the variance of the linear density field including modes up to
$\Lambda$ as $\sigma^{2}(\Lambda)$, i.e.

$\sigma^{2}_{\Lambda}=\int_{\bm{p}}^{\Lambda}P_{\rm L}(p).$ (1.21)

#### Feynman rules.

  * •

The different propagators are denoted via

$\displaystyle\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_Plin.pdf}}}\qquad\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_Plin_cut.pdf}}}\qquad\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_Plin_shell.pdf}}}.$

  * •

There are two types of vertices: large boxes representing the operator
convolution Eq. (1.5) and small boxes indicating the expansion of $\delta$ in
terms of linear legs Eq. (1.11), i.e.,

$\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_opvertex.pdf}}}\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_Fvertex.pdf}}}\quad.$
(1.22)

Notice that the linear legs are always represented by dashed lines, to make
the distinction between both vertices clearer.

##  2 RG flow via the partition function

In this section we compute the RG running of the bias parameters using the
partition function formalism. The partition function for matter can be written
as [3, 20]

$\displaystyle\mathcal{Z}^{\rm m}[J_{\Lambda}]$
$\displaystyle=\int\mathcal{D}\delta^{(1)}_{\Lambda}\mathcal{P}[\delta^{(1)}_{\Lambda}]\exp\left(\int_{\bm{k}}J_{\Lambda}(\bm{k})\delta_{\rm
fwd}[\delta^{(1)}_{\Lambda}](-\bm{k})\right)\,,\,$ (2.1)

where $\mathcal{P}[\delta^{(1)}_{\Lambda}]$ is defined in Eq. (1.15) and
$\delta_{\rm fwd}[\delta^{(1)}_{\Lambda}]$ denotes the perturbative (forward-
evolved) solution to the equations of motion for the matter density field. As
discussed in Sec. 1, we have to insert the solution of the equations of motion
in the coupling to the source current $J_{\Lambda}$, which, as indicated by
the subscript, only has support over wavenumbers below $\Lambda$. Moreover, we
only keep the fastest-growing mode, allowing us to reduce the initial
conditions to a single field $\delta^{(1)}$. Notice that we already normalized
the partition function in $\mathcal{P}[\delta^{(1)}_{\Lambda}]$ [see Eq.
(1.15)], i.e. $\mathcal{Z}^{\rm m}[0]=1$.

The partition function generates all $n$-point correlation functions of the
matter density field via functional derivatives with respect to $J_{\Lambda}$
[3]. The finite cutoff ensures that all observables derived from $\mathcal{Z}$
are finite, but of course accurate only below $\Lambda$, which is why we
consider a source current cut at $\Lambda$. However, since the cutoff is
imposed artificially as a calculational tool, observables should not depend on
$\Lambda$. In order to satisfy this requirement in general, one has to
introduce counterterms whose $\Lambda$-dependent coefficients absorb the
cutoff dependence. Physically, the counterterms capture the effect of small-
scale modes that have been integrated out to obtain Eq. (2.1). In the case of
the matter density, the leading such counterterm is
$c_{\nabla^{2}\delta}(\Lambda)\nabla^{2}\delta$ [1].

We now turn to the corresponding partition function for biased tracers. It can
be similarly guessed from the form of the bias expansion Eq. (1.1) to be [20]

$\displaystyle\mathcal{Z}[J_{\Lambda}]=\int\mathcal{D}\delta^{(1)}_{\Lambda}\mathcal{P}[\delta^{(1)}_{\Lambda}]\exp$
$\displaystyle\left(\int_{\bm{k}}J_{\Lambda}(\bm{k})\left[\sum_{O}b_{O}^{\Lambda}O[\delta^{(1)}_{\Lambda}](-\bm{k})\right]\right.$
$\displaystyle\quad\left.+\frac{1}{2}P_{\epsilon}^{\Lambda}\int_{\bm{k}}J_{\Lambda}(\bm{k})J_{\Lambda}(-\bm{k})+\mathcal{O}[J_{\Lambda}^{2}\delta^{(1)}_{\Lambda},\
J_{\Lambda}^{3}]\right)\,.$ (2.2)

There are thus two differences to Eq. (2.1). First, we have included the
counterterms, including their $\Lambda$-dependent bias coefficients
$b_{O}^{\Lambda}$, since they already appear at linear order and lowest order
in derivatives for biased tracers. Second, we have a term $\propto
J_{\Lambda}^{2}$, which captures stochasticity (noise) in the tracer field
[20]. Specifically, we include the leading Gaussian part
$P_{\epsilon}^{\Lambda}\equiv P_{\epsilon}^{\\{0\\},\Lambda}$ [cf. Eq.
(1.14)]; this is proportional to the variance of the field
$\epsilon(\bm{x},\tau)$ in Eq. (1.1)]. On the other hand, we have not written
the stochastic terms that incorporate the coupling with the operators
$O[\delta^{(1)}_{\Lambda}]$, corresponding to the $c_{\epsilon,O}$ in Eq.
(1.1). We also have not explicitly written non-Gaussian stochastic terms which
are of order $J_{\Lambda}^{3}$ and higher [30], relevant, for instance, for
the bispectrum. We will neglect these terms throughout the paper as they are
not relevant for the running of the bias parameters. Their investigation is
left for future work. Note that a noise contribution exists also for the
matter density, which we have not written in Eq. (2.1), as it is suppressed by
$k^{4}$, and hence only relevant at fairly high order in perturbation theory.

Similarly to Eq. (2.1), Eq. (2.2) generates all $n$-point correlation
functions of the galaxy density field $\delta_{g}$. In addition, it can be
used to derive the probability density functional (or “likelihood”) of the
tracer field [20].

###  2.1 General RG equations

We start by reviewing the results from [3], who derived the running of the
matter counterterms using the partition function. They expand the action in
Eq. (2.1) as a general series in $\delta^{(1)}$ and $J$, introducing index
notation for momentum vectors and Einstein summation convention:

$S_{\rm
int}[\delta^{(1)}_{\Lambda},J^{\Lambda}]=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}\frac{1}{n!m!}K^{j_{1}\ldots
j_{n}}_{i_{1}\ldots
i_{m}}(\Lambda)(\delta^{(1)}_{\Lambda})^{i_{1}}\cdots(\delta^{(1)}_{\Lambda})^{i_{m}}\;J^{\Lambda}_{j_{1}}\ldots
J^{\Lambda}_{j_{n}}.$ (2.3)

We have already assumed only the fastest growing mode, allowing us to reduce
the degrees of freedom from $\phi=(\delta,\theta)$ to $\delta$. Here, the
terms with $n=1$ correspond to the “deterministic” (mean-field) prediction for
$\delta_{g}$, i.e. the first line of Eq. (2.2), while contributions with
$n>1$, i.e. those nonlinear in the current $J_{\Lambda}$, encode stochasticity
as noted above. Each of the kernels $K$ is a sum of several terms involving
contributions from several operators at different orders in perturbations. In
fact, this expansion of the effective action and the RG equations derived from
it are directly applicable to the case of biased tracers [Eq. (2.2)] as well.
Thus, the kernels include the bias and stochastic parameters from the large-
scale bias expansion, but defined at the scale $\Lambda$.

Ref. [3] then derive RG equations for the coefficients $K^{j_{1}\ldots
j_{n}}_{i_{1}\ldots i_{m}}(\Lambda)$ [Eq. (4.22) there]. For a kernel of order
$n$ (upper indices), the source terms in the RG equation only involve orders
$\leq n$ (this justifies why we can drop non-Gaussian stochastic terms in this
paper). Since kernels with $n=0$ vanish (for Gaussian initial conditions, at
least), the RG equations for the $n=1$ (bias) and $n=2$ (Gaussian stochastic)
kernels become

$\displaystyle\frac{d}{d\Lambda}K^{j}_{i_{1}\ldots i_{m}}$
$\displaystyle=-\frac{1}{2}\frac{d(P_{\rm
L}^{\Lambda})^{kl}}{d\Lambda}K^{j}_{kli_{1}\ldots i_{m}}$ (2.4)
$\displaystyle\frac{d}{d\Lambda}K^{j_{1}j_{2}}_{i_{1}\ldots i_{m}}$
$\displaystyle=-\frac{1}{2}\frac{d(P_{\rm
L}^{\Lambda})^{kl}}{d\Lambda}K^{j_{1}j_{2}}_{kli_{1}\ldots
i_{m}}+2\frac{d(P_{\rm L}^{\Lambda})^{kl}}{d\Lambda}\sum_{k=0}^{m}{m\choose
k}K^{j_{1}}_{ki_{1}\cdots i_{k}}K^{j_{2}}_{li_{k+1}i_{m}},$ (2.5)

where $d(P_{\rm L}^{\Lambda})^{kl}/d\Lambda=\delta_{\rm K}^{kl}P_{\rm shell}$
is the shell propagator written in index notation. Note that, if and only if
the bias expansion is complete at each order in perturbations, the kernels $K$
depend on $\Lambda$ only through the coefficients $b_{O}^{\Lambda}$. In the
next section, we will derive the RG equations explicitly at the level of bias
operators, show that indeed the $b_{O}^{\Lambda}$ completely absorbs the
running of the kernels, and connect the results with the general RG equation
above.

###  2.2 RG equations for biased tracers

The partition function $\mathcal{Z}[J]$ defined at some scale
$\Lambda^{\prime}$ predicts the running of the bias coefficients
$b_{O}^{\Lambda^{\prime}}$ to any lower scale $\Lambda<\Lambda^{\prime}$. To
compute this, we consider the running over an infinitesimal distance from
$\Lambda^{\prime}$ to $\Lambda=\Lambda^{\prime}-\lambda$. This corresponds to
integrating out modes in a shell of width $\lambda$ in momentum space. In RG
flow computations, it is common to do this by computing the derivative of
$\mathcal{Z}$ with respect to $\Lambda$; here we begin by considering a finite
cutoff shift $\lambda$, which is assumed to be infinitesimal afterwards.

The computation of the running proceeds by integrating out the modes of the
linear density field within the shell $|\bm{k}|\in(\Lambda,\Lambda^{\prime}]$.
Correspondingly, we separate $\delta^{(1)}_{\Lambda^{\prime}}$ into two parts:

$\delta^{(1)}_{\Lambda^{\prime}}(\bm{k})=\delta^{(1)}_{\Lambda}(\bm{k})+\delta^{(1)}_{\rm
shell}(\bm{k}),$ (2.6)

where $\delta^{(1)}_{\rm shell}(\bm{k})$ has support
$|\bm{k}|\in(\Lambda,\Lambda^{\prime}]$, and its power spectrum $P_{\rm
shell}$ is defined in Eq. (1.19). The path integral in Eq. (2.2) factorizes,
leading to

$\displaystyle\mathcal{Z}[J_{\Lambda}]=\int\mathcal{D}\delta^{(1)}_{\Lambda}\mathcal{P}[\delta^{(1)}_{\Lambda}]\int\mathcal{D}\delta^{(1)}_{\rm
shell}\mathcal{P}[\delta^{(1)}_{\rm shell}]$ (2.7)
$\displaystyle\quad\times\exp\left(\int_{\bm{k}}J_{\Lambda}(\bm{k})\left[\sum_{O}b_{O}^{\Lambda^{\prime}}O[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](-\bm{k})\right]+\frac{1}{2}P_{\epsilon}^{\Lambda^{\prime}}\int_{\bm{k}}J_{\Lambda}(\bm{k})J_{\Lambda}(-\bm{k})+\mathcal{O}[J_{\Lambda}^{2}\delta^{(1)}_{\Lambda},\
J_{\Lambda}^{3}]\right)\,.$

We emphasize again that $J_{\Lambda}$ has support only at scales below
$\Lambda$, while the right-hand side integrates over all modes
$\delta^{(1)}_{\Lambda^{\prime}}=\delta^{(1)}_{\Lambda+\lambda}$. Thus,
$J_{\Lambda}$ has no support in the shell, which guarantees that the product
$P_{\rm shell}(k)J_{\Lambda}(k)$ is zero (in other words, $J_{\Lambda}$ and
$\delta^{(1)}_{\rm shell}$ are orthogonal). If we are interested in computing
observables up to a maximum scale $k_{\rm max}$, we can in fact choose $J$ to
have support only up to $k_{\rm max}$, and choosing
$\Lambda^{\prime}>\Lambda>k_{\rm max}$ is sufficient to fulfill this condition
(where all of these scales are below $k_{\rm NL}$). In perturbation theory,
every operator $O$ is evaluated up to some finite maximum order. Let us
consider the $n$-th order contribution $O^{(n)}$, which involves $n$ powers of
the linear density field (including of course the contributions from the
nonlinear evolution of matter itself). Thus, we can write

$\displaystyle O^{(n)}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm shell}]$
$\displaystyle=$ $\displaystyle
O^{(n)}[\delta^{(1)}_{\Lambda}]+O^{(n),{(1)_{\rm
shell}}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm shell}]+O^{(n),{(2)_{\rm
shell}}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm shell}]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\ldots+O^{(n),{(n-1)_{\rm
shell}}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}]+O^{(n)}[\delta^{(1)}_{\rm shell}],$

where the $O^{(n),{(m)_{\rm shell}}}$ involve $m$ powers of $\delta^{(1)}_{\rm
shell}$ and $n-m$ powers of $\delta^{(1)}_{\Lambda}$. Trivially in this
notation $O^{(n),{(0)_{\rm shell}}}=O^{(n)}[\delta^{(1)}_{\Lambda}]$ and
$O^{(n),{(n)_{\rm shell}}}=O^{(n)}[\delta^{(1)}_{\rm shell}]$. We refer the
reader to App. A.1 for examples on how this expansion works for the basis of
operators considered in this paper.

Note that in the limit $\lambda\to 0$, $\delta^{(1)}_{\rm shell}$ is
parametrically smaller than $\delta^{(1)}_{\Lambda}$; to see this, note that
$\langle\delta^{(1)}_{\rm
shell}(\bm{x})^{2}\rangle=(d\sigma_{\Lambda}^{2}/d\Lambda)\lambda$ [see Eq.
(2.18) below], which in the limit $\lambda\to 0$ is much smaller than
$\langle\delta^{(1)}_{\Lambda}(\bm{x})^{2}\rangle=\sigma_{\Lambda}^{2}$. Using
this, we can expand the exponential in Eq. (2.7) in powers of
$\delta^{(1)}_{\rm shell}$:

$\displaystyle\mathcal{Z}[J_{\Lambda}]$
$\displaystyle=\int\mathcal{D}\delta^{(1)}_{\Lambda}\mathcal{P}[\delta^{(1)}_{\Lambda}]\exp\left(\int_{\bm{k}}J_{\Lambda}(\bm{k})\left[\sum_{O}b_{O}^{\Lambda^{\prime}}O[\delta^{(1)}_{\Lambda}](-\bm{k})\right]\right.$
$\displaystyle\qquad\qquad\left.+\frac{1}{2}P_{\epsilon}^{\Lambda^{\prime}}\int_{\bm{k}}J_{\Lambda}(\bm{k})J_{\Lambda}(-\bm{k})+\mathcal{O}[J_{\Lambda}^{2}\delta^{(1)}_{\Lambda},\
J_{\Lambda}^{3}]\right)$
$\displaystyle\qquad\times\bigg{(}1+\int_{\bm{k}}J_{\Lambda}(\bm{k})\left[\sum_{O}b_{O}^{\Lambda^{\prime}}\left(\mathcal{S}_{O}^{1}[\delta^{(1)}_{\Lambda}](-\bm{k})+\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}](-\bm{k})+\ldots\right)\right]$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{\bm{k},\bm{k}^{\prime}}J_{\Lambda}(\bm{k})J_{\Lambda}(\bm{k}^{\prime})\sum_{O,O^{\prime}}b_{O}^{\Lambda^{\prime}}b_{O^{\prime}}^{\Lambda^{\prime}}\left[\mathcal{S}_{OO^{\prime}}^{11}[\delta^{(1)}_{\Lambda}](\bm{k},\bm{k}^{\prime})+\ldots\right]+\mathcal{O}[J_{\Lambda}^{2}\delta^{(1)}_{\Lambda},\
J_{\Lambda}^{3}]\bigg{)},$ (2.9)

where we introduce the operator expectation values

$\displaystyle\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]$
$\displaystyle=\sum_{n\geq 2}\int\mathcal{D}\delta^{(1)}_{\rm
shell}\mathcal{P}[\delta^{(1)}_{\rm shell}]\>O^{(n),(2)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm shell}](\bm{k})$
$\displaystyle=\sum_{n\geq 2}\left\langle O^{(n),(2)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k})\right\rangle_{\rm shell}$ (2.10)
$\displaystyle\mathcal{S}_{OO^{\prime}}^{11}[\delta^{(1)}_{\Lambda}](\bm{k},\bm{k}^{\prime})$
$\displaystyle=\sum_{n,n^{\prime}\geq 1}\int\mathcal{D}\delta^{(1)}_{\rm
shell}\mathcal{P}[\delta^{(1)}_{\rm shell}]\>O^{(n),(1)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k})\>O^{\prime(n^{\prime}),(1)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm shell}](\bm{k}^{\prime})$
$\displaystyle=\sum_{n,n^{\prime}\geq 1}\left\langle O^{(n),(1)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k})\>O^{\prime(n^{\prime}),(1)_{\rm
shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k}^{\prime})\right\rangle_{\rm shell}\,,$

where the sums run over the contributions to $O$ and $O^{\prime}$ in
perturbation theory. On the other hand,

$\mathcal{S}_{O}^{1}[\delta^{(1)}_{\Lambda}]=\sum_{n\geq
1}\int\mathcal{D}\delta^{(1)}_{\rm shell}\mathcal{P}[\delta^{(1)}_{\rm
shell}]\>O^{(n),(1)_{\rm shell}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k})=0$ (2.11)

vanishes, since the integrand is linear in $\delta^{(1)}_{\rm shell}$ by
definition. We also emphasize that these expectation values are functionals of
$\delta^{(1)}_{\Lambda}$; that is, the long-wavelength modes are not
integrated out. Terms that are proportional to more than two powers of
$\delta^{(1)}_{\rm shell}$, represented in Eq. (2.2) by dots, are suppressed
such that we drop them hereafter. Thus, only $S_{O}^{2}$ and
$S_{OO^{\prime}}^{11}$ need to be considered.

We can then reconstruct the exponential from its series and compare it to the
partition function for the bias and stochastic parameters at the scale
$\Lambda$:

$\displaystyle\mathcal{Z}[J_{\Lambda}]$
$\displaystyle=\int\mathcal{D}\delta^{(1)}_{\Lambda}\mathcal{P}[\delta^{(1)}_{\Lambda}]\exp\left[\int_{\bm{k}}J_{\Lambda}(\bm{k})\left(\sum_{O}b_{O}^{\Lambda^{\prime}}\left[O[\delta^{(1)}_{\Lambda}](-\bm{k})+\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}](-\bm{k})\right]\right)\right.$
(2.12)
$\displaystyle\left.\qquad+\frac{1}{2}P_{\epsilon}^{\Lambda^{\prime}}\int_{\bm{k},\bm{k}^{\prime}}J_{\Lambda}(\bm{k})J_{\Lambda}(\bm{k}^{\prime})\left((2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})+\sum_{O,O^{\prime}}b_{O}^{\Lambda^{\prime}}b_{O^{\prime}}^{\Lambda^{\prime}}\mathcal{S}_{OO^{\prime}}^{11}[\delta^{(1)}_{\Lambda}](\bm{k},\bm{k}^{\prime})\right)\right]\,.$

Fig. 1 provides a diagrammatic representation for the shell correlators
$\mathcal{S}_{O}^{2}$ (top) and $\mathcal{S}_{OO^{\prime}}^{11}$ (bottom). We
can also connect these results to the general RG equations in Eqs.
(2.4)–(2.5), by noting that $\mathcal{S}_{O}^{2}(\bm{k}_{1},\ldots\bm{k}_{m})$
corresponds to the contribution of the operator $O$ to

$\frac{d(P_{\rm L}^{\Lambda})^{kl}}{d\Lambda}K^{j}_{kli_{1}\ldots
i_{m}}\Big{|}_{O}\,,$ (2.13)

and correspondingly
$\mathcal{S}_{OO^{\prime}}^{11}(\bm{k}_{1},\ldots\bm{k}_{m})$ is the
contribution of operators $O,O^{\prime}$ to

$\frac{d(P_{\rm L}^{\Lambda})^{kl}}{d\Lambda}K^{j_{1}}_{ki_{1}\cdots
i_{k}}\Big{|}_{O}K^{j_{2}}_{li_{k+1}i_{m}}\Big{|}_{O^{\prime}}\,.$ (2.14)

Figure 1: Diagrams representing the shell correlators $\mathcal{S}_{O}^{2}$
(top) and $\mathcal{S}_{OO^{\prime}}^{11}$ (bottom).

Note that $S_{OO^{\prime}}^{11}$ multiplies two powers of the current
$J_{\Lambda}$. This means that it contributes to the stochastic part of the
partition function, in particular to the coupling between
$\mathcal{O}(J_{\Lambda}^{2})$ with long-wavelength operators
$O[\delta^{(1)}_{\Lambda}]$. Since we disregard these higher-order stochastic
contributions in this paper, we do not need to consider $S_{OO^{\prime}}^{11}$
further. We note though that it is ultimately these stochastic terms which
also produce the leading stochastic term
$P_{\epsilon}^{\Lambda}\int_{\bm{k}}J_{\Lambda}(\bm{k})J_{\Lambda}(-\bm{k})$
under the RG flow.

Let us then focus on the terms linear in $J_{\Lambda}$. By comparing Eq.
(2.12) with the partition function Eq. (2.2) at the scale $\Lambda$, we can
read off

$\sum_{O}b_{O}^{\Lambda}O[\delta^{(1)}_{\Lambda}]=\sum_{O}b_{O}^{\Lambda^{\prime}}\left(O[\delta^{(1)}_{\Lambda}]+\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]\right).$
(2.15)

If (and only if) the set of bias operators is closed under renormalization,
the shell expectation value $\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]$ can
depend on $\delta^{(1)}_{\Lambda}$ only through the bias operators themselves.
That is, we should be able to write

$\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]=\lambda\sum_{O^{\prime}}\Theta_{O,O^{\prime}}(\Lambda)O^{\prime}[\delta^{(1)}_{\Lambda}],$
(2.16)

where we have used the fact that $\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]$
is first order in the shell width $\lambda$. We can anticipate that

$\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}]\propto\int_{\bm{p}}P_{\rm
shell}(p)\,,$ (2.17)

and we will show this explicitly in the next section. Since [cf. Eq. (1.21)]

$\int_{\bm{p}}P_{\rm
shell}(p)=\int_{\Lambda}^{\Lambda+\lambda}\frac{p^{2}dp}{2\pi^{2}}P_{\rm
L}(p)=\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\Big{|}_{\Lambda}\lambda+\mathcal{O}(\lambda^{2})\,,$
(2.18)

we can isolate the leading $\Lambda$ dependence in $\Theta$ as

$\Theta_{O,O^{\prime}}(\Lambda^{\prime})=c_{O,O^{\prime}}(\Lambda)\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\Big{|}_{\Lambda}\,,$
(2.19)

where $c_{O,O^{\prime}}(\Lambda)$ are coefficients which will be determined
below. For operators $O$ that are leading-order in derivatives,
$c_{O,O^{\prime}}$ turn out to be $\Lambda$-independent coefficients of order
unity. For higher-derivative operators $O$, they are $\Lambda$-dependent, with
the powers of $\Lambda$ determined by the dimensions of the operator.

Inserting Eq. (2.16) into Eq. (2.15), and reordering the sum, we obtain

$\sum_{O}b_{O}^{\Lambda}O[\delta^{(1)}_{\Lambda}]=\sum_{O}\left[b_{O}^{\Lambda^{\prime}}+\lambda\sum_{O^{\prime}}\Theta_{O^{\prime}O}(\Lambda)b_{O^{\prime}}^{\Lambda^{\prime}}\right]O[\delta^{(1)}_{\Lambda}]\,.$
(2.20)

Now we can identify coefficients and read off

$b_{O}^{\Lambda}=b_{O}^{\Lambda^{\prime}}+\lambda\sum_{O^{\prime}}\Theta_{O^{\prime}O}(\Lambda)b_{O^{\prime}}^{\Lambda^{\prime}}\,,$
(2.21)

which via Eq. (2.19) and since $\Lambda=\Lambda^{\prime}-\lambda$ becomes

$\frac{d}{d\Lambda}b_{O}^{\Lambda}=-\frac{d}{d\lambda}b_{O}^{\Lambda}=-\sum_{O^{\prime}}\Theta_{O^{\prime}O}(\Lambda)b_{O^{\prime}}^{\Lambda}=-\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\sum_{O^{\prime}}c_{O^{\prime}O}\,b_{O^{\prime}}^{\Lambda}\,.$
(2.22)

In order to derive the RG equations for the $b_{O}^{\Lambda}$, we thus only
need to determine the coefficients $c_{O^{\prime}O}$.

###  2.3 Running of the bias parameters

We now compute the running of the bias parameters, evaluating the ensemble
average of correlators over the initial modes in the shell described by Eq.
(2.10), keeping the modes $|\bm{k}|<\Lambda$ fixed. This is a central
difference to the $n$-point-function-based renormalization from [11]: our
approach keeps all linear modes below the cutoff explicit, whereas the
$n$-point-function-based renormalization marginalizes over all modes,
including large scales.

We evaluate the running of linear and second-order bias parameters considering
up to third-order corrections to those operators. As argued above, the only
non-trivial contributions are the $\mathcal{S}^{2}_{O}$ terms

$\displaystyle\mathcal{S}_{O}^{2}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle\langle O^{{(2)_{\rm
shell}}}[\delta^{(1)}_{\Lambda},\delta^{(1)}_{\rm
shell}](\bm{k}_{1})\rangle_{\rm shell}\,,\,$ (2.23)

with contributions $\mathcal{S}_{O}^{\geq 3}$ being suppressed by
$\delta^{(1)}_{\rm shell}$. We now proceed to calculate those shell ensemble
averages for operators that are up to the third order in the bias expansion,
with the goal of computing the coefficients in the expansion Eq. (2.16). We
refer to App. A.2 for the complete calculation, and highlight in the main text
the most relevant results. We also comment later on how fourth-order operators
affect the RG running of second-order bias operators.

The shell integrals for $\mathcal{S}^{2}_{\delta}$ leads to terms that are
proportional to $\int_{\bm{p}}F_{\ell}(\bm{p},-\bm{p},\dots)P_{\rm shell}(p)$;
specifically,

$\displaystyle\mathcal{S}^{2}_{\delta}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
3\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}F_{3}(\bm{p},-\bm{p},\bm{k})P_{\rm
shell}(p)$ (2.24)
$\displaystyle\quad+\,6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{4}(\bm{p},-\bm{p},\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\quad+\,\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
$\displaystyle\stackrel{{\scriptstyle k\ll p}}{{=}}$
$\displaystyle-\frac{61}{315}k^{2}\delta^{(1)}_{\Lambda}(\bm{k})\int\frac{p^{2}dp}{2\pi^{2}}\,\frac{P_{\rm
shell}(p)}{p^{2}}+\,\mathcal{O}\left[\nabla^{2}\left(\delta^{(1)}_{\Lambda}\right)^{2}\right]\,,$

where $\mathcal{O}[\nabla^{2}(\delta^{(1)}_{\Lambda})^{2}]$ represents higher-
derivative contributions that are also higher order in
$\delta^{(1)}_{\Lambda}$. After integrating out modes in the shell, the linear
operator $\delta$ thus only contributes a correction to the $\nabla^{2}\delta$
operator. Diagrammatically, higher-derivative operators appear from shell
contractions on the same $F_{\ell}$ kernel, represented by the right top panel
of Fig. 1. This is due to the “double-softness” of the $F_{\ell}$ kernels,
inherited from mass and momentum conservation [31]. In the case of $\delta$,
this is the only type of contraction that exists, leading to the conclusion
that $\delta$ can only source higher-derivative terms.

The shell contraction for the operator $\delta^{2}$ leads to

$\displaystyle\mathcal{S}^{2}_{\delta^{2}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=4\,\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}F_{2}(\bm{k},\bm{p})P_{\rm
shell}(p)$ (2.25)
$\displaystyle+6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[F_{3}(\bm{p},-\bm{p},\bm{p}_{1})+F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})\right]P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,.$

Here and throughout we assumed $\bm{k}\neq 0$, so that we can drop tadpole
contributions such as $(2\pi)^{3}\delta_{\rm D}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)$. The first term leads to a contribution to $\delta$ and the term
proportional to $F_{3}(\bm{p},-\bm{p},\bm{p}_{1})$ leads to corrections to
$\nabla^{2}\delta$. The terms proportional to
$F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})$ and
$F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})$ lead to corrections of the type
$(\bm{p}_{1}\cdot\bm{p}_{2})/(p_{1}^{2}p_{2}^{2})$, which correspond to non-
Galilean-invariant displacements $\partial_{i}\Phi_{g}\partial_{j}\Phi_{g}$.
When summed together, however, those contributions cancel out, similarly to
what was found by [11]. The sum of all contributions in Eq. (2.25) leads to

$\displaystyle\mathcal{S}^{2}_{\delta^{2}}[\delta^{(1)}_{\Lambda}](\bm{k})=$
$\displaystyle\left[\frac{68}{21}\delta^{(1+2)}(\bm{k})+\frac{8126}{2205}\left[\delta^{2}(\bm{k})\right]^{(2)}+\frac{254}{2205}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)$ $\displaystyle\quad+\mbox{higher derivative
(h.d.)}+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,,$
(2.26)

where we omitted higher-derivative (h.d.) and higher-order operators. Notice
that Eq. (2.3) has the same structure and coefficients as Eq. (2.44) of [11]
[once one converts $\mathcal{G}_{2}$ to
$(\partial_{i}\partial_{j}\Phi_{g})^{2}$ via Eq. (1.7)]222Although the
numerical coefficients are the same, the frameworks are essentially different.
Ref. [11] calculates those coefficients setting a renormalization condition in
the IR (see Sec. 3), which ultimately leads to contractions with linear legs.
In our framework, we integrate out a UV shell based on [3]. Differences appear
for instance via non-linear terms when including primordial non-Gaussianities
which will generate non-linear terms in the running.; the only difference is
that we integrate over an infinitesimal shell at finite cutoff $\Lambda$,
while [11] integrate over all modes up to $\Lambda$ and send
$\Lambda\to\infty$. Non-trivially, notice that the prefactor in front of
$\delta^{(1)}(\bm{k})$ and $\delta^{(2)}(\bm{k})$ are the same, despite the
fact that one comes from the first term in Eq. (2.25) and the other comes from
the sum of $F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})$ and
$F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})$. This is a direct consequence of the
Galilean invariance of the underlying physics: if $\delta^{(2)}$ received a
different contribution than $\delta^{(1)}$, then the running of $\Lambda$
would force us to introduce a new bias term for the displacement term
$s^{i}\partial_{i}\delta$ which is part of $\delta^{(2)}$. This term is not
Galilei-invariant however and should not appear in the bias expansion. Another
way to say this is to note that the bias coefficients at fixed scale $\Lambda$
are physical parameters, and so their running has to be independent of the
order in perturbation theory that we work in. In conclusion, we see that
$\mathcal{S}^{2}_{\delta^{2}}$ sources $\delta,\,\delta^{2}$ and
$\mathcal{G}_{2}$.

The $\delta^{3}$ contribution leads to

$\displaystyle\mathcal{S}^{2}_{\delta^{3}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
3\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+3\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+3\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
$\displaystyle=$ $\displaystyle
3\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(1+2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\frac{17}{7}\left(\delta^{2}[\delta_{\Lambda}^{(1)}]\right)^{(2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,,$

where we see that the first two terms contribute to
$\delta[\delta_{\Lambda}^{(1)}]$ (at leading and second order) and the final
term contributes to $\delta^{2}[\delta_{\Lambda}^{(1)}]$. Moreover, for
$\mathcal{G}_{2}\delta$ we have

$\displaystyle\mathcal{S}^{2}_{\mathcal{G}_{2}\delta}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$
$\displaystyle\left[-\frac{4}{3}\delta^{(1+2)}(\bm{k})-\frac{376}{105}\delta^{2,(2)}(\bm{k})+\frac{116}{105}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)$ (2.28)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\mbox{h.d.}+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,.$

Again, notice that (both for $\mathcal{S}^{2}_{\mathcal{G}_{2}\delta}$ and
$\mathcal{S}^{2}_{\delta^{3}}$) the contributions to
$\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(1)}$ and
$\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(2)}$ have the same prefactor.
The Galileon operators $\mathcal{G}_{2},\,\mathcal{G}_{3}$ and also
$\Gamma_{3}$ will only lead to corrections to higher-derivative operators.
This is a direct consequence of the non-renormalization property of the
Galileon operators [32, 33]. Finally, higher-derivative terms such as
$\nabla^{2}\delta$ can only source higher-derivative terms, due to the
external $k^{2}$ momenta. To include one example of fourth-order operators,
the correction sourced by the fourth-order $\delta^{4}$ operator is also easy
to calculate:

$\displaystyle\mathcal{S}^{2}_{\delta^{4}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle 6\int_{\bm{p},\bm{p}_{3}}P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{k}-\bm{p}_{3})+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
(2.29) $\displaystyle=$ $\displaystyle
6\left(\delta^{2}[\delta_{\Lambda}^{(1)}]\right)^{(2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,,$

which contributes only to $\delta^{2}[\delta_{\Lambda}^{(1)}]$ at the order we
work in.

Putting all the contributions together, we obtain the galaxy bias RG equations
directly from Eq. (2.22):

$\displaystyle\frac{db_{\delta}}{d\Lambda}$ $\displaystyle=$
$\displaystyle-\left[\frac{68}{21}b_{\delta^{2}}+3b_{\delta^{3}}^{\ast}-\frac{4}{3}b_{\mathcal{G}_{2}\delta}^{\ast}\right]\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\,,$
(2.30) $\displaystyle\frac{db_{\delta^{2}}}{d\Lambda}$ $\displaystyle=$
$\displaystyle-\left[\frac{8126}{2205}b_{\delta^{2}}+\frac{17}{7}b_{\delta^{3}}^{\ast}-\frac{376}{105}b_{\mathcal{G}_{2}\delta}^{\ast}+b_{n=4}^{\ast(\delta^{2})}\right]\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\,,$
(2.31) $\displaystyle\frac{db_{\mathcal{G}_{2}}}{d\Lambda}$ $\displaystyle=$
$\displaystyle-\left[\frac{254}{2205}b_{\delta^{2}}+\frac{116}{105}b_{\mathcal{G}_{2}\delta}^{\ast}+b_{n=4}^{\ast(\mathcal{G}_{2})}\right]\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\,.$
(2.32)

The system of equations above is the analogue of the $\beta$-function running
for QFT, also known as the Callan-Symanzik equation [18, 19]. We use the
notation $b_{n=4}^{\ast(\delta^{2})}=6b_{\delta^{4}}^{\ast}+\dots$ to account
for the fourth order contributions to $\delta^{2}$ and similarly for
$b_{n=4}^{\ast(\mathcal{G}_{2})}$, which, as noted above, we have not derived.
We use the superscript $b^{\ast}$ to indicate bias parameters that do not
evolve in $\Lambda$ and are fixed at some scale $\Lambda_{\ast}$. Note that we
already approximated the third and fourth-order bias parameters as constants;
computing the running of those parameters would demand calculating the shell
contractions for sixth-order operators. We comment on this approximation and
how to go beyond it below.

We also find that the sourced higher-derivative terms have a different running
compared to the other bias parameters, proportional to

$\int_{\bm{p}}\frac{P_{\rm
shell}(p)}{p^{2}}=\int_{\Lambda}^{\Lambda+\lambda}\frac{dp}{2\pi^{2}}P_{\rm
L}(p)\,.$ (2.33)

Higher-derivative terms only source higher-derivative terms, and hence do not
affect the running of the other bias parameters (while being sourced by them,
however).

###  2.4 Quantitative results and closure of RG hierarchy

We highlight that the RG equations form a linear ODE system333This is the case
solely for Gaussian initial conditions. Non-Gaussian initial conditions will
introduce nonlinear running of the bias and stochastic parameters [3]. for the
bias parameters in which, in general, the bias coefficients of order $n$
contribute to bias parameters at order $n-2$ and higher.

All dependencies on cosmological parameters of the RG-flow we have derived are
absorbed by $d\sigma^{2}_{\Lambda}/d\Lambda$. This is a consequence of the
fact that we have assumed the Einstein-de Sitter form of the perturbation-
theory kernels. Including the cosmology dependence in the kernels [29, 34]
would lead to very small, cosmology-dependent corrections to the coefficients
in the RG equations. In our case however, we can change the integration
variable to $\sigma^{2}$ to have cosmology-independent equations

$\displaystyle\frac{db_{\delta}}{d\sigma^{2}}$ $\displaystyle=$
$\displaystyle-\left[\frac{68}{21}b_{\delta^{2}}+b_{n=3}^{\ast(\delta)}\right]\,,$
(2.34) $\displaystyle\frac{db_{\delta^{2}}}{d\sigma^{2}}$ $\displaystyle=$
$\displaystyle-\left[\frac{8126}{2205}b_{\delta^{2}}+b_{n=3+4}^{\ast(\delta^{2})}\right]\,,$
(2.35) $\displaystyle\frac{db_{\mathcal{G}_{2}}}{d\sigma^{2}}$
$\displaystyle=$
$\displaystyle-\left[\frac{254}{2205}b_{\delta^{2}}+b_{n=3+4}^{\ast(\mathcal{G}_{2})}\right]\,,$
(2.36)

in which we also included the third-order bias
$\\{b_{\delta^{3}}^{\ast},b_{\mathcal{G}_{3}}^{\ast},b_{\mathcal{G}_{2}\delta}^{\ast},b_{\Gamma_{3}}^{\ast}\\}$
as (constant) source terms using the notation
$b_{n=3+4}^{\ast}=b_{n=3}^{\ast}+b_{n=4}^{\ast}$. Thus, the ODE system above
forms a complete set and the equation for $b_{\delta^{2}}$ decouples from the
others. Neglecting the running of higher-order bias coefficients, i.e. setting
them to constants, and for initial conditions
$\bm{b}^{\ast}=\bm{b}\,(\sigma^{2}=\sigma^{2}_{\ast})=\left(b_{\delta}^{\ast},b_{\delta^{2}}^{\ast},b_{\mathcal{G}_{2}}^{\ast}\right)$,
where $\sigma_{*}^{2}\equiv\sigma_{\Lambda_{*}}^{2}$, we find:

$\displaystyle b_{\delta}(\sigma^{2})$ $\displaystyle=$ $\displaystyle
b_{\delta}^{\ast}+b_{\delta^{2}}^{\ast}\frac{c_{\delta,\delta^{2}}}{c_{\delta^{2},\delta^{2}}}\left(e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}-1\right)$
$\displaystyle\quad-\,b_{n=3}^{\ast(\delta)}(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})-b_{n=3+4}^{\ast(\delta^{2})}\frac{c_{\delta,\delta^{2}}}{c_{\delta^{2},\delta^{2}}^{2}}\left((\text{$\sigma^{2}$}-\sigma^{2}_{*})c_{\delta^{2},\delta^{2}}+e^{-(\sigma^{2}-\text{$\sigma^{2}_{*}$})c_{\delta^{2},\delta^{2}}}-1\right)\,,$
$\displaystyle b_{\delta^{2}}(\sigma^{2})$ $\displaystyle=$ $\displaystyle
b_{\delta^{2}}^{\ast}e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}$
$\displaystyle\quad+b_{n=3+4}^{\ast(\delta^{2})}\left(\frac{e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}}{c_{\delta^{2},\delta^{2}}}-\frac{1}{c_{\delta^{2},\delta^{2}}}\right)\,,$
$\displaystyle b_{\mathcal{G}_{2}}(\sigma^{2})$ $\displaystyle=$
$\displaystyle
b_{\mathcal{G}_{2}}^{\ast}+b_{\delta^{2}}^{\ast}\frac{c_{\mathcal{G}_{2},\delta^{2}}}{c_{\delta^{2},\delta^{2}}}\left(e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}-1\right)$
$\displaystyle\quad-\,b_{n=3+4}^{\ast(\mathcal{G}_{2})}(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})-b_{n=3+4}^{\ast(\delta^{2})}\frac{c_{\mathcal{G}_{2},\delta^{2}}}{c_{\delta^{2},\delta^{2}}^{2}}\left((\text{$\sigma^{2}$}-\sigma^{2}_{\ast})c_{\delta^{2},\delta^{2}}+e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}-1\right)\,,$

with $c_{\delta,\delta^{2}}=28/61$, $c_{\delta^{2},\delta^{2}}=8126/2205$ and
$c_{\mathcal{G}_{2},\delta^{2}}=254/2205$. Notice that we are fixing the bias
parameters at a reference scale $\Lambda_{\ast}$. We discuss later the running
of the bias for different initial conditions.

To recap, we have derived the complete set of contributions from operators up
to including third order (and $\delta^{4}$) to operators up to second order.
This allows us to consistently compute the running of bias coefficients up to
second order, assuming third and higher orders are zero at the initial scale
$\Lambda_{*}$. If we include nonzero third-order operators at $\Lambda_{*}$,
we have to make assumptions about how they evolve with $\Lambda$. We will
illustrate these different cases below, and discuss how the bias RG equations
are truncated. Note the difference to classic RG flow examples in
renormalizable field theories, which only have a finite number of interaction
vertices: in the present case, and as usual in effective field theories, new
interaction vertices (bias operators) appear at each order. We split the
discussion of this system of equations and solutions into three scenarios:
first, by setting high-order operators to zero $b_{n\geq 3}^{\ast}=0$; second,
considering $b_{n=3}$ and $b_{n=4}$ as constants and third, we comment on the
expected behavior when taking into account the full set of dynamical ODE’s at
higher orders and how to truncate it. The first two scenarios are based
directly on the solutions above. Later we discuss solutions for the third
scenario.

Figure 2: Bias RG-flow for zero higher-order bias coefficients. The top panels
show the bias coefficients as a function of the cutoff $\Lambda$ for two
different initial conditions set at $\sigma^{2}_{\ast}=0.5$. The bottom panels
show the trajectories for different initial conditions in the
$b_{\delta^{2}},b_{\delta}$ (left) and $b_{\delta^{2}},b_{\mathcal{G}_{2}}$
(right) planes running towards $\Lambda\to 0$.

#### No higher-order bias ($b_{n=3+4}^{(O)}=0$).

Starting from the scenario in which higher-order operators are zero, we
display in the top panels of Fig. 2 the bias RG-flow as a function of
$\sigma^{2}$ and the cutoff $\Lambda$ (for which we assume a $\Lambda$CDM
cosmology [28]). As examples, we set two different initial conditions
$\bm{b}^{\ast}$ at $\sigma^{2}_{\ast}=0.5$:
$\left(b_{\delta}^{\ast},b_{\delta^{2}}^{\ast},b_{\mathcal{G}_{2}}^{\ast}\right)=\left(1,0.5,0\right)$
(top left) and
$\left(b_{\delta}^{\ast},b_{\delta^{2}}^{\ast},b_{\mathcal{G}_{2}}^{\ast}\right)=\left(1,-0.5,0\right)$
(top right). Note that in both cases $b_{\mathcal{G}_{2}}$ is generated by
$b_{\delta}^{2}$ despite setting it to zero in the initial conditions, which
is in tandem with the findings of [11]. It is also worth mentioning that
$b_{1}$ can change sign depending on the sign of $b_{\delta^{2}}$ in this
scenario. In the bottom panels of Fig. 2, the lines indicate the evolution of
the trajectories in the $b_{\delta^{2}},b_{\delta}$ (left) and
$b_{\delta^{2}},b_{\mathcal{G}_{2}}$ (right) planes when running towards
$\Lambda\to 0$ starting from different initial conditions. Note that the
running of $b_{\mathcal{G}_{2}}$ is substantially slower compared to the other
bias parameters. The stability analysis of the system indicates that the only
(unstable) fixed point is at $b_{\delta^{2}}=0$. We see from the top panel of
Fig. 2 that the bias parameters exponentially depend on the cutoff but
asymptote to a finite constant on large scales, determining the endpoints of
the lines seen in the bottom panels.444As an aside, note from Eqs. (2.4)-(2.4)
that, for the (non-perturbative) limit $\sigma^{2}\to\infty$ we find that
$b_{\delta^{2}}\to 0$ and $b_{\delta},b_{\mathcal{G}_{2}}$ approach a
constant. In this limit, however, second-order operators also source higher-
order operators that back-react and change this solution.

Figure 3: Same as Fig. 2, but with higher-order bias parameters as constant
sources. The left and right panels show two different initial conditions set
at $\sigma^{2}_{*}=0.5$.

#### Constant higher-order bias ($b_{n=3+4}^{(O)}=b_{n=3+4}^{\ast(O)}$).

In the second scenario, we include third and fourth-order terms as static
sources. The top panels of Fig. 3 show the running of the bias parameters for
fixed initial conditions at $\sigma^{2}_{\ast}=0.5$. The left panel represents
the scenario in which all higher-order parameters are order one. For that
case, we still see the sign change for the bias parameters (especially
dependent on the sign of $b_{\delta^{2}}$). In the right panel, we take
$b_{n=3+4}^{\ast(\delta^{2})}=10\gg 1$ as the only non-vanishing parameter. In
that case, the other bias parameters are generated by this source term. The
bottom panels indicate the trajectories in the $b_{\delta^{2}},b_{\delta}$
plane for those two different initial conditions for higher-order terms. When
the source becomes parametrically large compared to the bias parameters
(right), more complicated trajectories start to form. In the case the sources
are order one (left panel), the system behaves similarly to the unsourced
scenario, but we can notice significant quantitative changes. This is shown in
Fig. 4, where we compare the bias evolution without (solid lines) and with
constant higher-order bias parameters (dashed lines). There is a noticeable
difference between both scenarios even at low $\Lambda$, showing that bias
terms of order $n+2$ cannot be neglected in the calculation of the running of
$O^{[n]}$.

Figure 4: Comparison of the RG-running for $b_{\delta}$ (top left),
$b_{\delta^{2}}$ (top right) and $b_{\mathcal{G}_{2}}$ (bottom) starting from
$\Lambda_{*}=0$ ($\sigma^{2}_{\ast}=0$) to higher $\Lambda$. The solid lines
show the result in the absence of higher-order bias, while the dashed and
dotted lines show the result for constant and running, via Eq. (2.40), higher-
order bias parameters. We have fixed
$\left(b_{\delta}^{\ast},b_{\delta^{2}}^{\ast},b_{\mathcal{G}_{2}}^{\ast}\right)=\left(1,1,0\right)$
as initial conditions,
$b_{n=3}^{\ast(\delta)}=b_{n=3+4}^{\ast(\delta^{2})}=b_{n=3+4}^{\ast(\mathcal{G}_{2})}=1$
(for the dashed and dotted cases) and
$c^{(\delta)}=c^{(\delta^{2})}=c^{(\mathcal{G}_{2})}=1$ (dotted case). We see
that, at sufficiently low $\Lambda$, the running of the higher-order
coefficients becomes irrelevant, showing that the RG equations can be
truncated.

#### Running higher-order bias and truncation of RG hierarchy.

Finally, we discuss the truncation of the hierarchy of bias RG equations. As
mentioned before, the complete running of the second-order bias parameters
demands calculating the fourth-order bias terms and those, in turn, also
depend on the running of sixth-order bias parameters. One may then wonder if
there is a systematic way to truncate this ODE system. Here we argue that, in
order to calculate the complete evolution of operators of order $n$, it is
sufficient to truncate the series at order $n+2$, at least if one is
interested in scales much below the initial scale $\Lambda_{*}$. This means
that for the calculation of the RG-flow for second-order operators it is
sufficient to consider third and fourth-order operators as static sources
(i.e. not including their RG equations which involve bias parameters that are
of fifth and higher order). Proving that this holds at all orders is beyond
the scope of this paper, but since the bias expansion guarantees that (at
least for large scales) the system is perturbative, we take the heuristic
approach of considering

$\displaystyle
b_{n=3+4}^{(O)}(\sigma^{2})=b^{*(O)}_{n=3+4}\,e^{-c^{(O)}(\sigma^{2}-\sigma^{2}_{\ast})}$
(2.40)

as an ansatz for the higher-order bias evolution, resembling the solutions
found in Eqs. (2.4)-(2.4). The two free parameters $b^{*(O)}$ and $c^{(O)}$
for each operator $O$ are, respectively, a proxy for the higher-order bias
coefficients and the shell prefactors (analogous to
$c_{\delta,\delta^{2}},\,c_{\delta^{2},\delta^{2}}$ and
$c_{\mathcal{G}_{2},\delta^{2}}$). Notice that we do not expect a hierarchy in
these coefficients [see Eqs. (2.31)-(2.32)]. After considering an ansatz of
the type (2.40) for $b_{n=3}^{(\delta)}(\sigma^{2})$,
$b_{n=3+4}^{(\delta^{2})}(\sigma^{2})$ and
$b_{n=3+4}^{(\mathcal{G}_{2})}(\sigma^{2})$ we can find an analytical solution
for the running (see Appendix B), which we show as dotted lines in Fig. 4. We
find that the running of the third and fourth-order bias parameters do not
affect the running of second-order parameters for sufficiently large scales,
as long as their coefficients $b^{(O)}$ and $c^{(O)}$ are $\mathcal{O}(1)$.

To calculate the $\Lambda$ evolution of the second-order parameters it is
therefore sufficient to truncate the ODE system at second order, and to
include third-order and fourth-order terms as static sources. The parametrized
running described here could also be included as an estimate of the theory
error in the running.

###  2.5 Main conclusions

We now summarize the relevant conclusions from the bias running:

  * •

In general, an operator $O$ of order $n$ will contribute to operators of order
$n-2$ and higher, which is a direct consequence of the shell contractions in
$\mathcal{S}^{2}_{O}$. Therefore, the calculation up to third-order terms
presented here gives the complete running of $b_{\delta}$. In order to obtain
the full running of second-order operators, one would need to calculate
fourth-order corrections, which we leave for future work. Also note that an
$n$-th order operator in general sources operators that are higher than $n$-th
order, for example:

$\raisebox{6.25963pt}{
\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_2_to_3a.pdf}}}\raisebox{0.0pt}{
\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/diag_2_to_3b.pdf}}}\qquad.$
(2.41)

The ODE system is then vertically coupled: even though the system may start
with zero higher-order operators, those will be sourced and backreact
affecting the running of lower-order terms. This effect is, however,
perturbatively suppressed.

  * •

In spite of the RG-flow equations being coupled at all orders, we find strong
indications, though not a rigorous proof, that the relevant contributions for
the running of an operator of order $n$ come from operators that are up to
order $n+2$ (Fig. 4). Higher-order operators only indirectly couple to the
lower-order terms, and are suppressed on large scales (i.e., low $\Lambda$).
This points to a natural way to truncate the ODE system.

  * •

Individual contributions from one term in the shell integrals can source
operators that break Galilean invariance, see Eqs. (2.25), (A.12) and (A.13).
When summed, those contributions cancel out. Similar findings were reported
for the $\delta^{2}$ renormalization in [11].

  * •

In the expansion of $\mathcal{S}^{2}_{O^{\prime}}$ in operators $O$, the
contribution to an operator $O^{(n)}$ is the same for different perturbative
orders $n$ [see for example the prefactors of $\delta$ in Eqs. (2.3), (2.3)
and (2.28)]. This fact can be understood as a statement that the bias
parameters $b_{O}^{\Lambda}$ are physical and independent of the order in
perturbation theory that the running is calculated at. Ultimately, this also
is a consequence of Galilean symmetry, and implies that the Galilei- or boost-
invariant basis of operators [6, 7] remains complete. The RG flow could
correspondingly be used to check for completeness of the bias expansion at
higher orders.

  * •

Contractions of two lines going into the same $F_{\ell}$ kernel, represented
by the top right diagram in Fig. 1, source higher-derivative operators.

  * •

Shell contractions between two different bias operators source higher-order
and higher-derivative stochastic contributions. Therefore, a complete
description of the galaxy density field has to include those terms.

In practice, one would vary the scale $\Lambda$, either at fixed $k_{\rm max}$
or keeping $k_{\rm max}$ as a fixed fraction of $\Lambda$, to confirm that
parameters that should be $\Lambda$-independent, such as cosmological
parameters, are in fact so. One can also compare the inferred running of bias
coefficients with that predicted by the theory as derived here as a check of
the relevance of higher-order contributions. In any case, we expect the
perturbative treatment to break down once $\Lambda$ approaches $k_{\rm NL}$.

##  3 Connection to bias parameters defined via $\bm{n}$-point functions

The (so far) standard approach to comparing theory with data is to use
$n$-point correlation functions, where the theory prediction is usually
calculated semi-analytically as the ensemble mean of a given correlation
function in perturbation theory. These correlation functions have leading and
next-to-leading order contributions; the latter in general need to be
regularized. In this context, Ref. [11] derived renormalization conditions
[Eq. (1.12)] which determine what counterterms need to be subtracted from a
given bias operator in order to regularize loop integrals [e.g., Eq. (1.13)].
For this, they introduced a cutoff, but merely as a calculational tool; in
practice, this cutoff is usually sent to infinity when numerically evaluating
loop integrals, as they converge to finite values at low loop orders (at least
in $\Lambda$CDM). Note the key difference to the cutoff used so far in this
paper, which is kept finite and explicit throughout. According to the
Wilsonian prescription, and as we comment later, the observables (i.e. the
$n$-point functions) in the end have to be independent of this artificial
(finite or infinite) cutoff while the free coefficients (coupling constants
and masses, or bias parameters) in general depend on $\Lambda$. The
information absorbed in the bias coefficients is different in both approaches,
however, since the finite cutoff allows us to keep more perturbative modes in
the loop integrals which are removed in the standard renormalized bias scheme.

In this section, we derive the connection between the finite-scale operators,
where $O=O[\delta^{(1)}_{\Lambda}]$ is constructed from the filtered density
field at the scale $\Lambda$,555We emphasize once more that
$O=O[\delta^{(1)}_{\Lambda}]$ is a short notation for
$O=O[\delta[\delta^{(1)}_{\Lambda}]]$. and the renormalized bias operators
$\llbracket O\rrbracket$, following [11] (see also Sec. 2.10 of [5]). The
connection between both frameworks is done in the “infrared” limit $\Lambda\to
0$; precisely, it is given by the relation

$\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm fixed}}\langle
O[\delta^{(1)}_{\Lambda}](\bm{k})\llbracket
O^{\prime}\rrbracket(\bm{k}^{\prime})\rangle=\lim_{\Lambda\to 0;\ k/\Lambda\
{\rm fixed}}\langle
O[\delta^{(1)}_{\Lambda}](\bm{k})O^{\prime}[\delta^{(1)}_{\Lambda}](\bm{k}^{\prime})\rangle\,.$
(3.1)

The goal of this section is to prove this relation. The reason we look at the
infrared limit is that first, this is what the definition of renormalized
operators refers to, and second, the calculation becomes much more tractable
and transparent in this limit. Readers who mainly want to understand the
relevance of the choice between finite-scale and $n$-point renormalization
scheme for observables such as the galaxy power spectrum may skip directly to
Sec. 4.

The renormalized operators are defined via renormalization conditions at the
$n$-point correlator level; specifically, one demands that for all $m\geq 0$,

$\langle\delta^{(1)}(\bm{k}_{1})\cdots\delta^{(1)}(\bm{k}_{m})\llbracket
O\rrbracket(\bm{k})\rangle\stackrel{{\scriptstyle k_{i}\to
0}}{{\longrightarrow}}\langle\delta^{(1)}(\bm{k}_{1})\cdots\delta^{(1)}(\bm{k}_{m})O[\delta](\bm{k})\rangle_{\rm
LO},$ (3.2)

where the subscript LO denotes the leading-order contribution in perturbation
theory, which we will define below. To provide some context for this
definition, note that any $n$-th order renormalized bias operator can be
unambiguously defined using the tree-level $(n+1)$-point function on large
scales. For example, $\llbracket\delta\rrbracket$ can be defined by the tree
level power spectrum in the limit $k\to 0$, and similarly for
$\llbracket\delta^{2}\rrbracket$ and $\llbracket\mathcal{G}_{2}\rrbracket$ via
the large-scale bispectrum. The renormalization condition Eq. (3.2) only
becomes nontrivial when loop contributions are considered. They are enforced
via counterterms to the renormalized operators which ensure that the bias
parameters measured do not depend on small-scale non-perturbative modes. In
summary, Eq. (3.2) implies that $\llbracket O\rrbracket$ is constructed
following Eq. (1.5), but with additional counterterms at a given order in
perturbations [see Eq. (1.13) for an example]. We will not need the explicit
expressions of the counterterms to obtain the connection to the field-level
operators, but comment on them later in the section.

Our goal in this section is to show that the bias coefficients
$b_{O}^{\Lambda}$ agree with those measured using $n$-point functions,
$b_{O}^{\rm n-pt}$ on large scales, i.e. in the limit of $\Lambda$ and $k$
going to zero. Apart from the conceptual connection between both schemes, this
also allows for a practical conversion of bias parameters. Specifically, one
would measure $b_{O}^{\Lambda_{\ast}}$ at some scale $\Lambda_{\ast}$, and
then run these down to $\Lambda\to 0$ via the running derived in the previous
section. The resulting bias coefficients are then equivalent to the
coefficients $b_{O}^{\rm n-pt}$ of the renormalized operators.

The equivalence of $b_{O}^{\Lambda}$ and $b_{O}^{\rm n-pt}$ in the infrared
limit can be made precise by stating that the corresponding operators agree in
a certain limit:

$L_{OO^{\prime}}\equiv\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm fixed}}\langle
O[\delta^{(1)}_{\Lambda}](\bm{k})\llbracket
O^{\prime}\rrbracket(\bm{k}^{\prime})\rangle=R_{OO^{\prime}}\equiv\lim_{\Lambda\to
0;\ k/\Lambda\ {\rm fixed}}\langle
O[\delta^{(1)}_{\Lambda}](\bm{k})O^{\prime}[\delta^{(1)}_{\Lambda}](\bm{k}^{\prime})\rangle.$
(3.3)

Notice that we take the large-scale limit keeping the ratio $k/\Lambda<1$
fixed. This ensures that $O[\delta^{(1)}_{\Lambda}]$ is only evaluated at
$k<\Lambda$ while we lower the cutoff. Suppose now that Eq. (3.3) holds. Since
both the set $\\{\llbracket O\rrbracket\\}$ and
$\\{O[\delta^{(1)}_{\Lambda}]\\}$ form a complete basis at any given order
(and for any finite $\Lambda$), we can write for a given tracer $\delta_{g}$

$\delta_{g}=\sum_{O}b_{O}^{\Lambda}\,O[\delta^{(1)}_{\Lambda}]=\sum_{O^{\prime}}b_{O^{\prime}}^{\rm
n-pt}\,\llbracket O^{\prime}\rrbracket,$ (3.4)

where both sums run over the same complete set of bias operators up to some
arbitrary but fixed order. Multiplying both sides by
$O^{\prime\prime}[\delta^{(1)}_{\Lambda}]$ and taking the ensemble average, we
see that

$M_{OO^{\prime}}\equiv\sum_{O^{\prime\prime}}(R^{-1})_{OO^{\prime\prime}}L_{O^{\prime}O^{\prime\prime}}$
(3.5)

is the invertible transformation matrix between the two operator conventions
$\llbracket O\rrbracket$ and $O[\delta^{(1)}_{\Lambda}]$. The bias
coefficients correspondingly transform with $(M^{-1})^{\top}$. Eq. (3.3) thus
says that in the $\Lambda\to 0$ limit, the two bases agree, i.e.
$M_{OO^{\prime}}$ is the identity, and hence the bias coefficients $b_{O}$
likewise agree for any given tracer.

Before delving into the derivation of Eq. (3.3), it is useful to make the
following distinction for the contributions to such correlators:

  * •

Gaussian: If instances of $\delta(\bm{p}_{i})$ from $O$ and $O^{\prime}$ are
set to $\delta^{(1)}(\bm{p}_{i})$ when inserting Eq. (1.5) into the operator
correlators in Eq. (3.3), we call this a “Gaussian contribution.” When
correlating an $n$-th order operator with an $m$-th order one, this involves
integration over precisely $n+m$ momenta. Consequently, after removing
Gaussian counterterms [see Eq. (3.7)], this contribution vanishes if $n\neq
m$. If $n=m$, this contribution is of order $(P_{\rm L})^{n}$. We mostly
single out the Gaussian contributions as a “warmup” to the general case.

  * •

Irreducible: Consider the cross-correlation of two operators evaluated at the
same perturbative order $n$, $\langle
O^{(n)}(\bm{k})O^{\prime(n)}(\bm{k}^{\prime})\rangle$. We refer to
$n$-particle-irreducible diagrams contributing to this correlator (or simply
“irreducible contributions”) as those for which all $n$ linear propagators are
connected between $O$ and $O^{\prime}$. The name follows from the fact that
all $n$ linear propagators would need to be cut to separate the diagram into
two different diagrams. Note that, for any $O,O^{\prime}$ these contributions
exist at arbitrarily high orders in perturbation theory. We will show that
irreducible contributions on both sides of Eq. (3.3) agree.

  * •

Reducible: Consider cross-correlation of two operators $\langle
O^{(n)}(\bm{k})O^{\prime(m)}(\bm{k}^{\prime})\rangle$ of in general different
perturbative orders. This correlation involves $(n+m)/2$ linear propagators
and vanishes if $n+m$ is odd, as we consider Gaussian initial conditions. We
refer to $(n+m)/2$-particle-reducible contributions (or simply reducible
contribution) as those for which not all linear propagators are connected
between $O$ and $O^{\prime}$. Note that if $n\neq m$, all contributions to the
correlator are reducible. We will show that reducible contributions on both
sides of Eq. (3.3) either vanish or are suppressed in the limit $\Lambda\to
0$.

We will consider each class of contributions separately in the discussion
below.

###  3.1 Gaussian contribution

Let us begin with the Gaussian result for $R_{OO^{\prime}}$,

$\displaystyle R_{OO^{\prime}}\Big{|}_{\rm
Gaussian}\stackrel{{\scriptstyle\rm}}{{=}}\lim_{\Lambda\to 0;k/\Lambda\ {\rm
fixed}}\langle
O[\delta^{(1)}_{\Lambda}](\bm{k})O^{\prime}[\delta^{(1)}_{\Lambda}](\bm{k}^{\prime})\rangle_{\rm
Gaussian}$ $\displaystyle=\lim_{\Lambda\to 0;k/\Lambda\ {\rm
fixed}}\int_{\bm{p}_{1},\ldots,\bm{p}_{n}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1\ldots
n})S_{O}(\bm{p}_{1},\ldots,\bm{p}_{n})\int_{\bm{p}^{\prime}_{1},\ldots,\bm{p}^{\prime}_{m}}\delta_{\rm
D}(\bm{k}^{\prime}-\bm{p}^{\prime}_{1\ldots
m})S_{O^{\prime}}(\bm{p}^{\prime}_{1},\ldots,\bm{p}_{m}^{\prime})$
$\displaystyle\qquad\qquad\times\left\langle\delta_{\Lambda}^{(1)}(\bm{p}_{1})\ldots\delta_{\Lambda}^{(1)}(\bm{p}_{n})\delta_{\Lambda}^{(1)}(\bm{p}^{\prime}_{1})\ldots\delta_{\Lambda}^{(1)}(\bm{p}^{\prime}_{m})\right\rangle.$
(3.6)

We now require that only contractions among the $\bm{p}_{i}$ and
$\bm{p}^{\prime}_{j}$ contribute to this correlator. This can be ensured at
the operator level by subtracting what we call Gaussian counterterms. For the
first few Eulerian operators, these are

$\delta^{2}(\bm{x})\to\delta^{2}(\bm{x})-\langle\delta^{2}\rangle;\quad\delta^{3}(\bm{x})\to\delta^{3}(\bm{x})-3\langle\delta^{2}\rangle\delta(\bm{x});\quad\mathcal{G}_{2}\delta(\bm{x})\to\mathcal{G}_{2}\delta(\bm{x})+\frac{4}{3}\langle\delta^{2}\rangle\delta(\bm{x}),$
(3.7)

and so on (see top panels of Fig. 5), where all instances of $\delta$
precisely denote $\delta[\delta^{(1)}_{\Lambda}](\bm{x})$. These are in fact
the same contributions that appear when performing a bias expansion in
Lagrangian space using orthogonal polynomials [35, 36, 37]. Since they can be
calculated based completely on the assumption of Gaussian statistics of
$\delta$, we refer to these as Gaussian counterterms. Notably, the Gaussian
counterterms were also included in the halo bias measurements of [38, 39], who
correlated operators constructed at the field level with the halo density
field in simulations. Notice that we have not subtracted the Gaussian
counterterms in the calculation of the RG equations in the previous section.
Indeed, one can check from Eqs. (2.3)–(2.28) that including these counterterms
removes the lowest-order contributions there [as noted after Eq. (2.3), we did
not write the tadpole contribution for $\delta^{2}$].

Figure 5: Example Gaussian contributions to $L_{OO^{\prime}}$. The
contribution on the right is absorbed by a Gaussian counterterm (e.g.,
$\delta^{2}\to\delta^{2}-\langle\delta^{2}\rangle$ and
$\delta^{3}\to\delta^{3}-3\langle\delta^{2}\rangle\delta$). After this
subtraction, as argued in the text, the Gaussian contributions agree with the
corresponding contributions to $R_{OO^{\prime}}$, thanks to Eq. (3.9).

Then, after removing the Gaussian counterterms, Eq. (3.6) vanishes if $m\neq
n$. A nonzero contribution arises from expanding $\delta_{\Lambda}$ beyond
linear order, i.e. inserting $F_{\ell}$ kernels with corresponding additional
integrated momenta into Eq. (3.6). We will consider this contribution below.

Now we can turn to the corresponding result for $L_{OO^{\prime}}$,

$\displaystyle L_{OO^{\prime}}\Big{|}_{\rm Gaussian}=\lim_{\Lambda\to 0;\
k/\Lambda\ {\rm fixed}}\langle O[\delta^{(1)}_{\Lambda}](\bm{k})\llbracket
O^{\prime}\rrbracket(\bm{k}^{\prime})\rangle_{\rm Gaussian}$
$\displaystyle=\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\int_{\bm{p}_{1},\ldots,\bm{p}_{n}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1\ldots
n})S_{O}(\bm{p}_{1},\ldots,\bm{p}_{n})\langle\delta_{\Lambda}^{(1)}(\bm{p}_{1})\ldots\delta_{\Lambda}^{(1)}(\bm{p}_{n})\llbracket
O^{\prime}\rrbracket(\bm{k}^{\prime})\rangle$ $\displaystyle=\lim_{\Lambda\to
0;\ k/\Lambda\ {\rm fixed}}\int_{\bm{p}_{1},\ldots,\bm{p}_{n}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1\ldots
n})S_{O}(\bm{p}_{1},\ldots,\bm{p}_{n})\langle\delta_{\Lambda}^{(1)}(\bm{p}_{1})\ldots\delta_{\Lambda}^{(1)}(\bm{p}_{n})O^{\prime}[\delta](\bm{k}^{\prime})\rangle_{\rm
LO}.$ (3.8)

The final equality follows from the definition of the $n$-pt renormalized
operators in the large-scale limit, Eq. (3.2). In particular, this statement
implies that there are only contractions across the $\bm{p}_{i}$ and the
momenta hidden in $O^{\prime}[\delta]$. Thus, if $n=m$, these are the same
contractions as in Eq. (3.6).

It then follows that the only difference between Eq. (3.6) and Eq. (3.8) are
the filters involved: in Eq. (3.6), both $\bm{p}_{i}$ and
$\bm{p}^{\prime}_{j}$ are cut at $\Lambda$, while in Eq. (3.8) only the
$\bm{p}_{i}$ are cut at $\Lambda$. However, since we have argued that only
cross-contractions between $\bm{p}_{i}$ and $\bm{p}^{\prime}_{j}$ contribute,
and since

$\langle\delta_{\Lambda}^{(1)}(\bm{p})\delta_{\Lambda}^{(1)}(\bm{p}^{\prime})\rangle=\langle\delta_{\Lambda}^{(1)}(\bm{p})\delta^{(1)}(\bm{p}^{\prime})\rangle$
(3.9)

for sharp-$k$ filters, it follows that Eq. (3.8) equals Eq. (3.6) for the
Gaussian contributions.

###  3.2 Irreducible contributions

Consider a contribution to $R_{OO^{\prime}}$ that involves the cross-
correlation of two operators evaluated at the same order $n$,

$R_{OO^{\prime}}\supset\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm fixed}}\langle
O^{(n)}[\delta^{(1)}_{\Lambda}](\bm{k})O^{\prime(n)}[\delta^{(1)}_{\Lambda}](\bm{k}^{\prime})\rangle.$
(3.10)

Expanding this correlator into Feynman diagrams, we can identify $n$-particle-
irreducible diagrams, and $n$-particle-reducible diagrams: the latter can be
separated into two separate diagrams by cutting less than $n$ linear
propagators, while the former cannot. Here, we only consider the $n$-particle-
irreducible diagrams, which we refer to as “irreducible contributions” for
short. Examples are shown in Fig. 6. It is clear that in these diagrams, each
linear propagator is connected to both $O$ and $O^{\prime}$. In fact, this
holds for all irreducible contributions. Note that we have not specified the
perturbative order $n$ here; in fact, irreducible contributions exist at
arbitrarily high order.

Figure 6:  Example irreducible contributions to $L_{OO^{\prime}}$. As argued
in the text, they agree with the corresponding contributions to
$R_{OO^{\prime}}$, thanks to Eq. (3.9).

As an example, consider
$O^{(2)}=\delta^{(2)},O^{\prime(2)}=(\delta^{2})^{(2)}$. Then, dropping the
tadpole counterterm $\langle\delta^{2}\rangle$,

$\displaystyle R_{\delta\delta^{2}}\Big{|}_{\rm irred.}=$
$\displaystyle\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\langle\delta^{(2)}[\delta^{(1)}_{\Lambda}](\bm{k})(\delta^{2}[\delta^{(1)}_{\Lambda}])(\bm{k}^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\,(2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}2\int_{\bm{p}}F_{2}(\bm{p},\bm{k}-\bm{p})P_{\rm
L}^{\Lambda}(\bm{p})P_{\rm L}^{\Lambda}(\bm{k}-\bm{p}).$ (3.11)

Let us then consider the corresponding contribution to $L_{\delta\delta^{2}}$,

$\displaystyle L_{\delta\delta^{2}}\Big{|}_{\rm irred.}=$
$\displaystyle\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\langle\delta^{(2)}[\delta^{(1)}_{\Lambda}](\bm{k})\llbracket\delta^{2}\rrbracket(\bm{k}^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\int_{\bm{p}}F_{2}(\bm{p},\bm{k}-\bm{p})\langle\delta_{\Lambda}^{(1)}(\bm{p})\delta_{\Lambda}^{(1)}(\bm{k}-\bm{p})\llbracket\delta^{2}\rrbracket(\bm{k}^{\prime})$
$\displaystyle=$ $\displaystyle\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\int_{\bm{p}}F_{2}(\bm{p},\bm{k}-\bm{p})\langle\delta_{\Lambda}^{(1)}(\bm{p})\delta_{\Lambda}^{(1)}(\bm{k}-\bm{p})(\delta^{2})(\bm{k}^{\prime})\rangle_{\rm
LO},$ (3.12)

where we have used the renormalization conditions in the second line (note
that we are taking both $k$ and $\Lambda$ to zero). The non-vanishing LO
contribution evaluates to the same as Eq. (3.11), thanks again to Eq. (3.9).
Note that
$\langle\delta^{(1)}(\bm{k})(\delta^{2})^{(3)}(\bm{k}^{\prime})\rangle$ is not
a reducible contribution; we will turn to those below.

The opposite case, $O=(\delta^{2})^{(2)},O^{\prime}=\delta^{(2)}$, is simpler
and yields

$L_{\delta^{2}\delta}\Big{|}_{\rm irred.}=\lim_{\Lambda\to 0;\ k/\Lambda\ {\rm
fixed}}\langle\delta^{2}[\delta^{(1)}_{\Lambda}](\bm{k})\llbracket\delta\rrbracket(\bm{k}^{\prime})\rangle=\lim_{\Lambda\to
0;\ k/\Lambda\ {\rm
fixed}}\langle\delta^{2}[\delta^{(1)}_{\Lambda}](\bm{k})\delta^{(2)}(\bm{k}^{\prime})\rangle_{\rm
LO},$ (3.13)

which is immediately seen to be equal to Eq. (3.11).

The diagrams in Fig. 6 show that this reasoning straightforwardly extends to
higher orders. By construction, in any irreducible diagram, all ingoing
momenta to $O[\delta^{(1)}_{\Lambda}]$ as well as $\llbracket
O^{\prime}\rrbracket$ are cut at $\Lambda$, leading to the same result for
$L_{OO^{\prime}}$ and $R_{OO^{\prime}}$. This is completely analogous to the
reasoning in [23].

We conclude that Eq. (3.3) holds for all irreducible contributions.

###  3.3 Reducible contributions

We now turn to the reducible contributions, which are simply defined as all
contributions that are not irreducible. By definition, this means that, in a
correlator $\langle O^{[n]}(\bm{k})O^{\prime[m]}(\bm{k}^{\prime})\rangle$, at
least one linear propagator is connected only within $O$ or $O^{\prime}$, so
that the entire diagram is $(n+m)/2$-particle-reducible. Note that if $n\neq
m$, all contributions are reducible.

Let us then consider the reducible contributions to $L_{OO^{\prime}}$ and
$R_{OO^{\prime}}$. We can consider two types of reducible contributions, based
on whether a linear propagator is connected to $O[\delta^{(1)}_{\Lambda}]$ or
only to $\llbracket O^{\prime}\rrbracket$:

  1. 1.

If at least one of the outgoing lines of a given propagator is connected to
$O[\delta^{(1)}_{\Lambda}]$, then, as argued in [23], this loop involves the
cut propagator $P_{\rm L}^{\Lambda}$, and the momentum is restricted to
$\leq\Lambda$; for example:

(3.14)

  2. 2.

If both lines of a propagator are connected to $\llbracket O\rrbracket$, then
the loop momentum is not regularized; for example:

(3.15)

By construction, in the limit of external momentum going to 0, this term is
entirely absorbed by a counterterm in $\llbracket O\rrbracket$. In the case of
the diagram above, involving a second-order operator $O^{[2]}$, the
counterterm is proportional to $\delta$ and, again, completely absorbs this
contribution.

To summarize, the second type of reducible contribution vanishes, while the
first has a loop momentum that is limited to $\leq\Lambda$, leading to a
suppression factor of $\sim\sigma_{\Lambda}^{2}\sim\Lambda^{3}P_{\rm
L}(\Lambda)/2\pi^{2}$ on large scales. Thus, this correction becomes
negligible in the $\Lambda\to 0$ limit considered here. The reducible
contributions to $R_{OO^{\prime}}$ are all of the first type. We conclude that
all reducible contributions either vanish or are suppressed in the limit
indicated in Eq. (3.3), and thus the asymptotic equality holds also beyond
leading order.

To summarize, Eq. (3.3) holds at all orders in perturbation theory and the
connection between the two renormalization schemes is realized at $\Lambda\to
0$.

##  4 On loop contributions to $\bm{n}$-point functions in the two approaches

We have argued in the previous section that, in the limit $\Lambda\to 0$ (with
$k/\Lambda$ fixed), the bias coefficients $b_{O}^{\Lambda}$ run to values that
agree with the $n$-point renormalized bias coefficients $b_{O}^{\rm n-pt}$
used in current EFT analyses.666Provided that the Gaussian counterterms are
included, as in Eq. (3.7). At finite $\Lambda$, the $b_{O}^{\Lambda}$ assume
different values, and hence no longer agree with the $b_{O}^{\rm n-pt}$. This
difference at finite $\Lambda$ is due to the different definitions of
operators. In this section we discuss how the UV physics is absorbed by the
bias parameters in the two different schemes. As an example, we consider the
three $O=\delta^{2}$ contributions to the 1-loop galaxy power spectrum, which
we discuss in turn below.

The
$\bm{\boldsymbol{\big{\langle}}(\delta)^{(1)}(\delta^{2})^{(3)}\boldsymbol{\big{\rangle}}}$
term. The first contribution is

$\displaystyle P_{gg}^{\textrm{1-loop, std}}(k)\supset\>$ $\displaystyle
2b_{\delta}^{\rm n-pt}b_{\delta^{2}}^{\rm
n-pt}\left\langle\llbracket\delta\rrbracket^{(1)}(\bm{k})\llbracket\delta^{2}\rrbracket^{(3)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=\>$ $\displaystyle 4b_{\delta}^{\rm n-pt}b_{\delta^{2}}^{\rm
n-pt}\>\int_{p}F_{2}(\bm{p},\bm{k}-\bm{p})P_{\rm L}(p)P_{\rm
L}(k)+\mbox{counterterms}.$ (4.1)

In the standard, $n$-point-renormalized approach, the $k/p\to 0$ limit
contribution is completely absorbed by the counterterm
$(68/21)\sigma^{2}\delta$ to $\llbracket\delta^{2}\rrbracket$; in fact, this
is a special case of the diagram in Eq. (3.15). The subleading contribution in
the limit $k/p\to 0$ is correspondingly absorbed by a counterterm involving
$\nabla^{2}\delta$ (as long as this piece is also properly subtracted). That
is, Eq. (4.1) disappears completely out of the 1-loop galaxy power spectrum.

On the other hand, when using a finite cutoff, the loop contribution becomes

$\displaystyle P_{gg}^{\textrm{1-loop, finite-scale}}(k)$
$\displaystyle\supset
2b_{\delta}^{\Lambda}b_{\delta^{2}}^{\Lambda}\left\langle\delta^{(1)}_{\Lambda}(\bm{k})\left(\delta^{2}[\delta^{(1)}_{\Lambda}]\right)^{(3)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=4b_{\delta}^{\Lambda}b_{\delta^{2}}^{\Lambda}\>\int_{p}F_{2}(\bm{p},\bm{k}-\bm{p})P_{\rm
L}^{\Lambda}(p)P_{\rm L}^{\Lambda}(k).$ (4.2)

This contribution [in fact a special case of Eq. (3.14)] is finite and is kept
as nonzero contribution to the 1-loop galaxy power spectrum. This explains
how, at finite $\Lambda$, a difference between $b_{\delta^{2}}^{\Lambda}$ and
$b_{\delta^{2}}^{\rm n-pt}$ arises. A corresponding finite contribution
$\propto(b_{\delta}^{\Lambda})^{2}b_{\delta^{2}}^{\Lambda}$ appears in the
galaxy bispectrum at 1-loop order,

$\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={1}]{figs/diags/BiLoopT1a.pdf}}}\,,$
(4.3)

which is controlled by the same $b_{\delta^{2}}^{\Lambda}$ as Eq. (4.2).

The
$\bm{\boldsymbol{\big{\langle}}(\delta^{2})^{(2)}(\delta^{2})^{(2)}\boldsymbol{\big{\rangle}}}$
term. The second contribution we consider is

$\displaystyle P_{gg}^{\textrm{1-loop, std}}(k)$
$\displaystyle\supset(b_{\delta^{2}}^{\rm
n-pt})^{2}\left\langle\llbracket\delta^{2}\rrbracket^{(2)}(\bm{k}^{\prime})\llbracket\delta^{2}\rrbracket^{(2)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=2(b_{\delta^{2}}^{\rm n-pt})^{2}\>\int_{\bm{p}}P_{\rm
L}(p)P_{\rm L}(|\bm{k}-\bm{p}|)+\mbox{counterterms}.$ (4.4)

This asymptotes to a UV-sensitive constant in the limit $k/p\to 0$, which is
thus to be subtracted, i.e. by removing the shot-noise-like counterterm
$\int_{\bm{p}}[P_{\rm L}(p)]^{2}$ from the integral [11]. The subleading
corrections scale analytically, i.e. as $k^{2}$, $k^{4}$, and so on. In order
to absorb the UV-dependence of the loop integral at subleading order, these
are thus to be correspondingly subtracted and absorbed by higher-derivative
stochastic counterterms $P_{\epsilon}^{\\{2\\},\rm
n-pt}k^{2},P_{\epsilon}^{\\{4\\},\rm n-pt}k^{4},...$ [cf. Eq. (1.14)]. The
corresponding contribution from
$\langle\llbracket\mathcal{G}_{2}\rrbracket^{(2)}(\bm{k}^{\prime})\llbracket\delta^{2}\rrbracket^{(2)}(\bm{k}^{\prime})\rangle$
vanishes in the large-scale limit, while the same argument as for
$\llbracket\delta^{2}\rrbracket$ holds for the subleading contributions.

On the other hand,

$\displaystyle P_{gg}^{\textrm{1-loop, finite-scale}}(k)$
$\displaystyle\supset(b_{\delta^{2}}^{\Lambda})^{2}\left\langle(\delta^{2}[\delta^{(1)}_{\Lambda}])^{(2)}(\bm{k}^{\prime})(\delta^{2}[\delta^{(1)}_{\Lambda}])^{(2)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=2\,(b_{\delta^{2}}^{\Lambda})^{2}\>\int_{\bm{p}}P_{\rm
L}^{\Lambda}(p)P_{\rm L}^{\Lambda}(|\bm{k}-\bm{p}|)$ (4.5)

is finite and kept in the finite-$\Lambda$ renormalization, as in Eq. (4.2).
In an analysis combining the galaxy power spectrum and bispectrum, the
constraint on $b_{\delta^{2}}$ from the tree-level bispectrum would then
constrain the finite contribution in Eq. (4.5). The $\Lambda$-dependence of
this term is still nonzero however, and needs to be absorbed by the effective
shot noise amplitude $P_{\epsilon}^{\Lambda}$.

The
$\bm{\boldsymbol{\big{\langle}}\delta^{(2)}(\delta^{2})^{(2)}\boldsymbol{\big{\rangle}}}$
term. The third and final contribution is

$\displaystyle P_{gg}^{\textrm{1-loop, std}}(k)$ $\displaystyle\supset
2b_{\delta}^{\rm n-pt}b_{\delta^{2}}^{\rm
n-pt}\left\langle\llbracket\delta\rrbracket^{(2)}(\bm{k})\llbracket\delta^{2}\rrbracket^{(2)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=4b_{\delta}^{\rm n-pt}b_{\delta^{2}}^{\rm
n-pt}\>\int_{p}F_{2}(\bm{p},\bm{k}-\bm{p})P_{\rm L}(p)P_{\rm
L}(|\bm{k}-\bm{p}|)+\mbox{counterterms},$ (4.6)

which in the limit $k/p\to 0$ scales as $k^{2}$, i.e. analytically. Thus, this
contribution is UV-sensitive and strictly also to be subtracted by a
subleading stochastic counterterm $P_{\epsilon}^{(2)}k^{2}$.

On the other hand,

$\displaystyle P_{gg}^{\textrm{1-loop, finite-scale}}(k)$
$\displaystyle\supset
2b_{\delta}^{\Lambda}b_{\delta^{2}}^{\Lambda}\left\langle\delta[\delta^{(1)}_{\Lambda}]^{(2)}(\bm{k})(\delta^{2}[\delta^{(1)}_{\Lambda}])^{(2)}(\bm{k}^{\prime})\right\rangle^{\prime}$
$\displaystyle=4b_{\delta}^{\Lambda}b_{\delta^{2}}^{\Lambda}\>\int_{p}F_{2}(\bm{p},\bm{k}-\bm{p})P_{\rm
L}^{\Lambda}(p)P_{\rm L}^{\Lambda}(|\bm{k}-\bm{p}|),$ (4.7)

which is again finite and now requires no further comment.

To summarize, the first point to emphasize is that, when implemented
consistently, both the usual, $n$-point renormalized bias expansion and the
finite-cutoff expansion at $\Lambda$ yield proper UV-insensitive predictions
that are independent of non-perturbative small-scale modes. However, in the
standard approach, the subtraction of the entire loop terms in Eq. (4.1), Eq.
(4.4), and Eq. (4.6) removes significant contributions from large-scale modes
that are in fact under control. This is particularly evident in the subleading
contributions in the $k/p\to 0$ limit, for example

$k^{2}P_{\rm L}(k)\int_{\bm{p}}p^{-2}P_{\rm
L}(p)\quad\mbox{[Eq.~{}(\ref{eq:dd2loop})];}\quad
k^{2}\int_{\bm{p}}p^{-2}[P_{\rm
L}(p)]^{2}\quad\mbox{[Eq.~{}(\ref{eq:d2d2loop})]}.$ (4.8)

As discussed above, these UV-sensitive terms should strictly be removed from
the loop integrals, and contribute via the counterterms
$b_{\nabla^{2}\delta}^{\rm n-pt}$, $P_{\epsilon}^{\\{2\\},\rm n-pt}$ instead.
However, they are large in the sense that they are much larger than $k_{\rm
NL}^{-2}P_{\rm L}(k)$ and $k_{\rm NL}^{-5}$, which would be the dimensional-
analysis estimate of the magnitude of UV contributions to each type of
counterterms, respectively. Hence, since the corresponding coefficients
$b_{\nabla^{2}\delta}^{\rm n-pt}$ and $P_{\epsilon}^{\\{2\\},\rm n-pt}$ now
include a significant contribution from low-$k$ modes, one expects that their
values are significantly enhanced. This would also mean that the length scale
controlling even higher-order higher-derivative contributions is likewise
enhanced.

Working with a finite cutoff, on the other hand, allows for keeping the large-
scale contributions in these loop integrals in a principled way. The
additional information in these particular loop terms was recently emphasized
in Sec. IV B of [40], who considered the covariance of $P_{gg}$ as well.
Fundamentally, the reason that these contributions are absorbed in the
standard approach is due to the fact that renormalization is defined via the
$k\to 0$ limit of $n$-point functions, which by construction marginalizes over
all low-$k$ modes.

At this point, however, we should emphasize that in current EFT analyses the
UV-sensitive contributions in Eq. (4.8) are usually not subtracted. This is
possible since, while in principle UV-sensitive, they converge to finite
values for a $\Lambda$CDM linear power spectrum. The general philosophy
appears to have been to remove only numerically divergent loop contributions
[such as those proportional to $\int_{\bm{p}}^{\infty}P_{\rm L}(p)$], while
retaining finite contributions such as the subleading contribution to Eq.
(4.4) [although the term in Eq. (4.1) is usually completely dropped, including
numerically finite subleading contributions]. This procedure is fine as long
as all counterterm coefficients, i.e. bias and stochastic parameters, are left
completely free. As soon as additional information is put on these parameters
in the form of priors, great care needs to be taken. E.g., the subleading
contribution to Eq. (4.4) is controlled by $(b_{\delta^{2}}^{\rm n-pt})^{2}$,
but its UV-sensitivity has to be absorbed by $P_{\epsilon}^{\\{2\\},\rm
n-pt}$. This means that these two parameters are correlated. Priors on the
latter parameter that do not consistently take into account this UV
sensitivity could bias the inferred value of $b_{\delta^{2}}^{\rm n-pt}$.
Similar arguments apply to higher-order statistics, such as the 1-loop galaxy
bispectrum, however involving now a substantially larger bias and stochastic
parameter space and even more complex interdependencies.

To reiterate, while priors for a given set of bias and stochastic parameters
are usually derived from physical arguments which don’t make reference to the
details of the UV-sensitive implementation of loop integrals, this is not
consistent with the renormalization scheme as currently employed, as these
parameters in fact have to absorb UV-sensitive loop contributions, which
moreover correlate different parameters. The non-negligible effect of priors
in EFTofLSS analyses has indeed been the focus of recent work [41, 42, 43].

To summarize, the proper $n$-point-function-baed renormalization demands a
meticulous (and error-prone) treatment of the basis of operators and loop
integrals. For the one-loop bispectrum, for example, it involves subtracting
many terms from fourth-order operators. Subleading analytic and higher-
derivative contributions (i.e. those scaling as $k^{2}/p^{2}$) have to be
removed from loop integrals. We emphasize again that all of these subtractions
are unnecessary, and all UV sensitivity is absent in the finite-scale
renormalization scheme presented in this work. The importance of consistent
and physically motivated priors on bias parameters (now at a given scale
$\Lambda_{*}$) applies equally in this case, however.

##  5 Discussion and conclusion

In this work, we have constructed the renormalization group equations for the
galaxy bias and stochastic parameters, Eqs. (2.30)-(2.32). We used the
partition function formalism of [3], generalized to include galaxy bias and
stochasticity following [20] to derive the running of those parameters as a
function of the smoothing scale $\Lambda$, as modes above $\Lambda$ in the
initial conditions are integrated out. We have discussed solutions to this ODE
system, indicating that in order to solve the RG evolution at a given order
$n$ in the bias expansion, one has to include bias operators up to order
$n+2$. We provide solutions for a given set of initial conditions and comment
on the stability of the joint first- and second-order bias parameter space
under RG flow. We also point out how the running of the bias parameters
generates higher-order stochastic contributions. A more detailed investigation
of stochasticity is relegated to future work.

We then connect the finite-cutoff bias parameters $b_{O}^{\Lambda}$ to their
counterparts $b_{O^{\prime}}^{\rm n-pt}$ defined using the standard $n$-point-
function-based renormalization procedure [11]. Specifically, we show that the
bias parameters of both schemes coincide in the limit $\Lambda\to 0$. The
finite cutoff bias parameters, however, allow us to predict additional
perturbative contributions to the galaxy $n$-point functions which so far have
usually been absorbed, partially or completely, via counterterms (see Sec. 4),
in particular into the effective shot noise amplitudes
$P_{\epsilon}^{\\{j\\},\rm n-pt}$. This indicates that the finite-cutoff
renormalization scheme in principle allows to extract more information from
the clustering of biased tracers than has been used in EFT-based analyses so
far (see also [40]). This information is available both at the field level and
at the level of summary statistics, as we comment on in Sec. 4 using the
contribution of $b_{\delta^{2}}^{\Lambda}$ to the power spectrum (at 1-loop
level) and the bispectrum (at tree level) as an example. A key difference to
the standard, $n$-point-function-based renormalization is that the
renormalization done here keeps all linear modes below the cutoff explicit;
subsequently, they can be either marginalized over (for $n$-point functions)
or determined by the data (in case of field-level inference). In contrast, the
$n$-point-function-based renormalization is done after marginalizing over all
modes, including large scales.

The difference in the effective shot noise treatment in the standard
renormalization approach and the bias expansion at a finite scale (see Sec. 4)
can have a particularly large impact for tracers with low “intrinsic” shot
noise [44, 45]. Intriguingly, the effective low-$k$ shot noise covariance
matrix of dark matter halos appears to have a large eigenvalue associated with
$b_{\delta^{2}}$ [46]; it is possible that this contribution can be predicted
perturbatively when adopting the finite-scale bias expansion. The actual
information gained in the latter over the standard approach will thus depend
on the details of the tracer sample, volume, and data vector chosen. This is
clearly an issue worthy of further investigation.

Finally, while the finite-scale bias expansion is currently mostly used in
field-level forward models [23, 10, 47], it is worth noting that it can
equally be used in standard semi-analytical calculations of $n$-point
functions. The presence of the cutoff will however complicate the use of
numerical techniques such as FFTlog [48]. This deserves further investigation,
as does the choice of smoother cutoff functions than the Heaviside form
adopted here (though a compact support in momentum space is essential to
ensure UV insensitivity).

## Acknowledgments

HR is supported by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy EXC 2094 ‘ORIGINS’. (No. 390783311). We thank Ivana Babić,
Giovanni Cabass, Stephen Chen, Mathias Garny, Julia Stadler, Beatriz Tucci and
Rodrigo Voivodic for discussions, and Ivana Babić, Julia Stadler and Mark Wise
for feedback on the paper.

##  Appendix A Shell integrals

In this appendix, we provide a complete calculation of the shell operators and
integrals from Sec. 2.

###  A.1 Shell operators

We display the shell expansion of the operators defined in Eq. (2.2) up to
third order in perturbation theory. Notice that only the contributions that
have exactly two powers $\delta^{(1)}_{\rm shell}$ will contribute to
$\mathcal{S}^{2}$ and therefore to the bias running. However, for
completeness, we display the full expressions for the shell expansion.

  * •

$O=\delta$:

$\displaystyle\delta[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\delta[\delta^{(1)}_{\Lambda}](\bm{k})+\delta^{(1)}_{\rm
shell}(\bm{k})$ (A.1)
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p}_{2})\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})\,F_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$
$\displaystyle\quad\times\left[3\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+3\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})\,F_{4}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4})\left[4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+6\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+\left.\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]\,.$

  * •

$O=\delta^{2}$:

$\displaystyle\delta^{2}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\delta^{2}[\delta^{(1)}_{\Lambda}](\bm{k})+\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})F_{2}(\bm{p}_{1},\bm{p}_{2})\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right.$
$\displaystyle\quad+\left.2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{2}(\bm{p}_{1},\bm{p}_{2})F_{2}(\bm{p}_{3},\bm{p}_{4})\left[4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+2\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad\left.+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})\left[3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+3\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+3\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+\left.\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})\right]$ (A.2)

  * •

$O=\delta^{3}$:

$\displaystyle\delta^{3}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\delta^{3}[\delta^{(1)}_{\Lambda}](\bm{k})+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})$ $\displaystyle\quad\times\left[3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+3\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{2}(\bm{p}_{1},\bm{p}_{2})\left[2\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+4\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ (A.3)
$\displaystyle\quad+2\,\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+2\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ $\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$

  * •

$O=\delta^{4}$:

$\displaystyle\delta^{4}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\delta^{4}[\delta^{(1)}_{\Lambda}](\bm{k})+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})\left[4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+6\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$ (A.4)
$\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]\,,$

  * •

$O=\mathcal{G}_{2}$:

$\displaystyle\mathcal{G}_{2}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\mathcal{G}_{2}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})F_{2}(\bm{p}_{1},\bm{p}_{2})\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2},\bm{p}_{3}}$
$\displaystyle\quad\times\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right.$
$\displaystyle\quad+\left.2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{2}(\bm{p}_{1},\bm{p}_{2})F_{2}(\bm{p}_{3},\bm{p}_{4})\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2},\bm{p}_{3}+\bm{p}_{4}}$
$\displaystyle\quad\left[4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+2\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad\left.+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3},\bm{p}_{4}}\left[3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+3\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$ (A.5)

  * •

$O=\mathcal{G}_{2}\delta$:

$\displaystyle\mathcal{G}_{2}\delta[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\mathcal{G}_{2}\delta[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right.$ $\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})+\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}F_{2}(\bm{p}_{3},\bm{p}_{4})\left[2\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+4\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$
$\displaystyle\quad+2\,\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+2\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ $\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})F_{2}(\bm{p}_{1},\bm{p}_{2})\sigma^{2}_{\bm{p}_{12},\bm{p}_{3}}\left[2\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ $\displaystyle\quad+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad+2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})$
$\displaystyle\quad+2\,\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})+\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ (A.6) $\displaystyle\quad\left.+\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$

  * •

$O=\mathcal{G}_{3}$. We do not display the shell opetaror for the third order
Galileon Eq. (1.8), which is rather long. The Galileon operators are non-
renormalizable [49] and therefore do not act as sources for other operators
[11].

  * •

$O=\Gamma_{3}$:

$\displaystyle\Gamma_{3}[\delta^{(1)}_{\Lambda}+\delta^{(1)}_{\rm
shell}](\bm{k})=\Gamma_{3}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\delta_{\rm
D}(\bm{k}-\bm{p}_{123})\left[F_{2}(\bm{p}_{1},\bm{p}_{2})-G_{2}(\bm{p}_{1},\bm{p}_{2})\right]\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2},\bm{p}_{3}}$
$\displaystyle\quad\times\left[2\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right.$
$\displaystyle\quad+\left.2\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\rm shell}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\right]$
$\displaystyle\quad+\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})\left[F_{2}(\bm{p}_{1},\bm{p}_{2})F_{2}(\bm{p}_{3},\bm{p}_{4})-G_{2}(\bm{p}_{1},\bm{p}_{2})G_{2}(\bm{p}_{3},\bm{p}_{4})\right]\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2},\bm{p}_{3}+\bm{p}_{4}}$
$\displaystyle\quad\times\left[4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\right.$
$\displaystyle\quad+2\,\delta^{(1)}_{\rm shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+4\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})$
$\displaystyle\quad\left.+\,\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$
$\displaystyle\quad+2\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\delta_{\rm
D}(\bm{k}-\bm{p}_{1234})\left[F_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})-G_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})\right]\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3},\bm{p}_{4}}$
$\displaystyle\quad\times\left[3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})\right.$ $\displaystyle\quad+3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm
shell}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+3\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\rm
shell}(\bm{p}_{4})$ (A.7) $\displaystyle\quad\left.+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})+\delta^{(1)}_{\rm
shell}(\bm{p}_{1})\delta^{(1)}_{\rm shell}(\bm{p}_{2})\delta^{(1)}_{\rm
shell}(\bm{p}_{3})\delta^{(1)}_{\rm shell}(\bm{p}_{4})\right]$

###  A.2 The $\mathcal{S}^{2}_{O}$ integrals

We now display the shell integrals $\mathcal{S}^{2}_{O}$ defined in Eqs.
(2.10) and (2.23), which correspond to taking the expectation value over
$\delta^{(1)}_{\rm shell}$ of the contributions
$\mathcal{O}([\delta^{(1)}_{\rm shell}]^{2})$ to the expansion defined in App.
A.1. As we discuss in Sec. 2, these expectation values will contribute to the
bias RG flow. Term by term, we find:

  * •

$O=\delta$:

$\displaystyle\mathcal{S}^{2}_{\delta}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
3\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}F_{3}(\bm{p},-\bm{p},\bm{k})P_{\rm
shell}(p)$ (A.8)
$\displaystyle\quad+\,6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{4}(\bm{p},-\bm{p},\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\quad+\,\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
$\displaystyle\stackrel{{\scriptstyle k\ll p}}{{=}}$
$\displaystyle-\frac{61}{315}k^{2}\delta^{(1)}_{\Lambda}(\bm{k})\int\frac{p^{2}dp}{2\pi^{2}}\,\frac{P_{\rm
shell}(p)}{p^{2}}+\,\mathcal{O}\left[\nabla^{2}\left(\delta^{(1)}_{\Lambda}\right)^{2}\right]\,,$

where $\mathcal{O}[\nabla^{2}(\delta^{(1)}_{\Lambda})^{2}]$ represents higher-
derivative contributions that are also higher order in
$\delta^{(1)}_{\Lambda}$. Notice that a higher-order expansion of $\delta$
will likewise only lead to contributions to higher-derivative operators, since
the momentum structure of the kernels is always of the form
$F_{\ell}(p,-p,\dots)$.

  * •

$O=\delta^{2}$:

$\displaystyle\mathcal{S}^{2}_{\delta^{2}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
4\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}F_{2}(\bm{k},\bm{p})P_{\rm
shell}(p)$ $\displaystyle+$ $\displaystyle\quad
6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[F_{3}(\bm{p},-\bm{p},\bm{p}_{1})+F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})\right]P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle\quad
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$
$\displaystyle\quad\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,.$

where we can immediately see that the first term will contribute to $\delta$

$\displaystyle
4\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}F_{2}(\bm{k},\bm{p})P_{\rm
shell}(p)$ $\displaystyle=$
$\displaystyle\frac{68}{21}\delta^{(1)}_{\Lambda}(\bm{k})\int\frac{p^{2}dp}{2\pi^{2}}\,P_{\rm
shell}(p)\,.$ (A.10)

One of the $F_{3}$ terms is also of the form $F_{n}(\bm{p},-\bm{p},\dots)$
such that

$\displaystyle\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{3}(\bm{p},-\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\propto$
(A.11) $\displaystyle\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})$ $\displaystyle\left(\frac{p_{1}}{p}\right)^{2}P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$

and it will renormalize $\delta\nabla^{2}\delta$. For the $22$ term, we find

$\displaystyle
4\int_{\bm{p}}F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})P_{\rm
shell}(p)=\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm shell}(p)\times$ (A.12)
$\displaystyle\quad\left[\frac{1916}{735}+\frac{32}{735}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})^{2}}{p_{1}^{2}p_{2}^{2}}-\frac{1}{3}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p^{2}}-\frac{1}{3}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{1}^{2}}-\frac{1}{3}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{2}^{2}}-\frac{1}{3}\frac{p^{2}(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{1}^{2}p_{2}^{2}}\right]\,.$

Notice that the first term will source contributions to $\delta^{2}$ and the
second to $\mathcal{G}_{2}$ and $\delta^{2}$. The third term in Eq. (A.12)
sources the higher-derivative term $\nabla_{i}\delta\,\nabla_{j}\delta$.
However, the last three terms in Eq. (A.12) do not respect Galilean
invariance, since they come from displacements terms like
$\partial_{i}\Phi\partial_{j}\Phi$ and $\partial_{i}\Phi\partial_{j}\delta$.
We comment more on them later. We now turn to
$F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})$, which we expand using $p\to\infty$ to
get

$\displaystyle
6\int_{\bm{p},\bm{p}_{1}}F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\stackrel{{\scriptstyle
p\to\infty}}{{\approx}}\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm shell}(p)\times$
(A.13)
$\displaystyle\quad\left[\frac{344}{105}+\frac{314}{315}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})^{2}}{p_{1}^{2}p_{2}^{2}}+\frac{41}{21}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{1}^{2}}+\frac{41}{21}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{2}^{2}}+\frac{1}{3}\frac{p^{2}(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{1}^{2}p_{2}^{2}}+\mathcal{O}\left(\frac{p_{1}^{2}}{p^{2}},\frac{p_{2}^{2}}{p^{2}}\right)\right]\,.$

We can see that the term proportional to
$(\bm{p}_{1}\cdot\bm{p}_{2})/(p_{1}^{2}p_{2}^{2})$ will will cancel out the
term from Eq. (A.12) and therefore the non-Galilean-invariant term
$\partial_{i}\Phi\partial_{j}\Phi$ disappears. We can then write

$\displaystyle\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}$ $\displaystyle\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[4F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})+6F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})\right]P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle=$ $\displaystyle\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\quad\times\left[\frac{4324}{735}+\frac{2294}{2205}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})^{2}}{p_{1}^{2}p_{2}^{2}}+\frac{34}{21}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{1}^{2}}+\frac{34}{21}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})}{p_{2}^{2}}\right]$
$\displaystyle=$
$\displaystyle\left[\frac{68}{21}\delta^{(2)}(\bm{k})+\frac{8126}{2205}(\delta^{2})^{(2)}(\bm{k})+\frac{254}{2205}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)\,,$ (A.14)

where we find that it will contribute to
$\delta,\,\delta^{2},\,\mathcal{G}_{2}$. Putting all terms together:

$\displaystyle\mathcal{S}^{2}_{\delta^{2}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$
$\displaystyle\left[\frac{68}{21}\delta^{(1+2)}(\bm{k})+\frac{8126}{2205}\left[\delta^{2}(\bm{k})\right]^{(2)}+\frac{254}{2205}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\,\mbox{higher
derivative
(h.d.)}+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,.$

An important result is that the coefficients of $\delta^{(1)}(\bm{k})$ and
$\delta^{(2)}(\bm{k})$ are the same, which can also be understood as a
consequence of Galilean invariance.

  * •

$O=\delta^{3}$:

$\displaystyle\mathcal{S}^{2}_{\delta^{3}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
3\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm