Resolving Muon g-2 Anomaly With Partial Compositeness

Shuai Xu1 and Sibo Zheng2

1School of Physics and Telecommunications Engineering, Zhoukou Normal
University, Henan 466001, China

2Department of Physics, Chongqing University, Chongqing 401331, China

###### Abstract

We consider the scenario of composite Higgs with partial compositeness to
address the muon g-2 anomaly. We show that this anomaly is resolved by one-
loop correction due to composite muon partners with mass scale of order TeVs
and large Yukawa coupling to composite Higgs, which is different from
interpretations of vectorlike lepton models. We present parameter space by
imposing indirect constraints from precise measurements on Higgs, Z and
oblique electroweak parameters. We analyze direct constraints in light of both
Drell-Yan and Higgs-associated productions of the composite muon partners at
high-luminosity LHC. It turns out that the parameter regions with the lighter
composite muon mass below $\sim 325$ GeV can be excluded at 2$\sigma$ order by
the Drell-Yan processes.

###### Contents

  1. 1 Introduction
  2. 2 Partial Compositeness
  3. 3 Muon g-2
    1. 3.1 Composite Quark Partners
    2. 3.2 Composite Lepton Partners
  4. 4 Constraints
    1. 4.1 Indirect Constraints
      1. 4.1.1 Higgs Portal
      2. 4.1.2 $Z$ Portal
      3. 4.1.3 Oblique Corrections
    2. 4.2 Direct Constraints
      1. 4.2.1 Gauge Boson Portal
      2. 4.2.2 Higgs Portal
  5. 5 Numerical Results
  6. 6 Conclusion

##  1 Introduction

The muon anomalous magnetic moment $a_{\mu}$ is an effective operator of
dimension five

$\displaystyle{}\mathcal{L}\supset\frac{e}{4m_{\mu}}a_{\mu}\bar{u}\sigma_{\rho\lambda}uF^{\rho\lambda},$
(1)

where $e$ is the electric charge, $m_{\mu}$ is the muon mass,
$\sigma^{\rho\lambda}=\frac{i}{2}[\gamma^{\rho},\gamma^{\lambda}]$, and
$F_{\rho\lambda}$ is the electromagnetic field strength. Measurement of this
observable provides a direct check on the completeness of a renormalizable
theory such as Standard Model (SM). This makes $a_{\mu}$ one of those
precisely calculated in the SM, see [1, 2] for reviews.

A comparison between the theoretical prediction of the SM and precise
measurements on $a_{\mu}$ enables us to uncover imprints of new physics. A so-
called muon anomaly has been reported by the BNL [3] experiment, which shows a
$3.7\sigma$ deviation of $a_{\mu}$ from the SM reference value.111The SM
theoretical value of $a_{\mu}$ is subject to an important uncertainty arising
from QCD hadron effects [5]. Recently, this deviation is further strengthened
by the latest FNAL [4] experiment with a larger significance of $4.2\sigma$.
Combining the BNL and FNAL data gives

$\displaystyle{}\delta a^{\rm{exp}}_{\mu}=(2.51\pm 0.59)\times 10^{-9}.$ (2)

To explain this anomaly, various models of new physics have been proposed in
the literature. Since $\delta a^{\rm{exp}}_{\mu}$ in Eq.(2) is of order one-
loop electroweak contribution to $a_{\mu}$, it is natural to consider new
electroweak degrees of freedom such as vectorlike leptons [6, 7, 8, 9, 10, 11,
12, 13] among others, see [14] for a comprehensive review on model buildings.
In this work, we consider an alternative scenario of composite Higgs with
partial compositeness, where composite muon partners play the role of the
vectorlike leptons. Previous work, which advocated interpreting the muon
anomaly with partial compositeness, can be found in [15, 16, 17, 18, 19]. In
this paper, we will consider this scenario by a systematic analysis, based on
a complete effective field theory description of this scenario. As in the
aforementioned vectorlike lepton models, the desired contribution to $a_{\mu}$
due to the composite lepton partners is expected to be constrained by their
effects on the other precise electroweak observables, either through gauge
and/or Yukawa interactions.

The rest of this paper is organized as follows. In Sec.2, we introduce the
effective field theory description of our scenario, with key features
different from those of vectorlike leptons emphasized. Sec.3 addresses the new
contributions to $a_{\mu}$ due to the composite quark and composite lepton
partners, which are two- and one-loop effect respectively. We will show that
the significance of the observed anomaly can be naturally reduced to 2$\sigma$
order by the composite muon partners with mass scale of order $\sim$ TeVs and
strong coupling constant $g_{*}$ of order unity. In Sec.4, we firstly consider
indirect constraints from precision measurements on Higgs, $Z$ and oblique
electroweak parameters, then discuss the prospect of searching for signals in
the parameter space in terms of both Drell-Yan and Higgs-associated
productions of the composite muon partners at the high-luminosity LHC. We
present the main numerical results in Sec.5. Finally, we conclude in Sec.6.

##  2 Partial Compositeness

In the framework of composite Higgs with partial compositeness, as firstly
proposed by refs.[20, 21], the general effective Lagrangian for partially
composite fermion degrees of freedom based on coset $\mathcal{G}/\mathcal{H}$
reads as

$\displaystyle{}\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\rm{e}}+\mathcal{L}_{\rm{s}}+\mathcal{L}_{\rm{m}}$
(3)

where the elementary sector, the strong sector, and the mixing effects are
respectively,

$\displaystyle{}\mathcal{L}_{\rm{e}}$ $\displaystyle=$ $\displaystyle
i\bar{q}_{L}\gamma^{\mu}D_{\mu}q_{L}+i\bar{u}_{R}\gamma^{\mu}D_{\mu}u_{R}+i\bar{d}_{R}\gamma^{\mu}D_{\mu}d_{R}+i\bar{\ell}\gamma^{\mu}D_{\mu}\ell+i\bar{e}_{R}\gamma^{\mu}D_{\mu}e_{R},$
$\displaystyle\mathcal{L}_{\rm{s}}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}D_{\mu}U(D^{\mu}U)^{{\dagger}}+\bar{\Psi}\left(i\gamma^{\mu}D_{\mu}-m_{\Psi}\right)\Psi$
$\displaystyle-$
$\displaystyle\left[g_{*}\left(Q_{L}\bar{U}_{R}H+Q_{L}\bar{D}_{R}\tilde{H}+L_{L}\bar{E}_{R}\tilde{H}\right)+g_{*}\left(Q_{R}\bar{U}_{L}H+Q_{R}\bar{D}_{L}\tilde{H}+L_{R}\bar{E}_{L}\tilde{H}\right)+h.c.\right]$
$\displaystyle+$ $\displaystyle\rm{higher-dimensional~{}terms},$
$\displaystyle\mathcal{L}_{\rm{m}}$ $\displaystyle=$
$\displaystyle-\sin\theta_{\ell_{L}}m_{L}\bar{\ell}_{L}L_{R}-\sin\theta_{\ell_{R}}m_{E}\bar{E}_{L}e_{R}+\cdots+\rm{h.c}.$
(4)

Here, $q_{L}$ and $\ell$ are the SM quark and lepton doublets respectively
whereas $u_{R}$, $d_{R}$ and $e_{R}$ are the $SU(2)_{L}$ singlets; with
respect to these SM states $Q$, $L$, $U$, $D$ and $E$ are the corresponding
vectorlike composite fermions that are components of representations of
$\mathcal{G}$; $H=(H^{+},\upsilon+h)^{T}$ is the composite Higgs doublet, a
pseudo Nambu-Goldstone boson described by the Goldstone matrix $U$ in Callan-
Coleman-Wess-Zumino formalism [22] with $\upsilon$ the weak scale and
$\tilde{H}=i\sigma_{2}H^{*}$; and $\Psi$ denotes any of the above composite
fermions. Note, besides the global group $\mathcal{G}/\mathcal{H}$ two
external groups, i.e, the QCD gauge group and $U(1)_{X}$ global group, are
also necessary, see [22] for a review.

The masses and couplings in Eq.(2) have the following physical contents, where
we will point out the key differences between this composite model and the
aforementioned vectorlike models.

  * •

The first term in $\mathcal{L}_{\rm{s}}$ is the kinetic term for the composite
Higgs with $f$ the $\mathcal{G}$-breaking mass scale, which includes both
derivative and non-derivative interactions involving the composite Higgs.
While these derivative interactions can help us distinguish a composite Higgs
model from vectorlike models, they are irrelevant for the topic of $a_{\mu}$
under consideration.

  * •

The second term in $\mathcal{L}_{\rm{s}}$ refers to the kinetic term for the
composite fermions with $m_{\Psi}$ their vectorlike masses, where the gauge
covariant derivative relies on explicit representations of the composite
states. Here, we simply adopt
$D_{\mu}=\partial_{\mu}-iYg^{\prime}B_{\mu}-igW_{\mu}$,222We will reminder the
reader when arguments related to this point are made. where $B_{\mu}$ is the
$U(1)_{Y}$ gauge field with gauge coupling $g^{\prime}$ and hypercharge $Y$,
and $W_{\mu}$ is the $SU(2)_{L}$ gauge field with gauge coupling $g$.

  * •

The last three terms in $\mathcal{L}_{\rm{s}}$ represent interactions between
the composite Higgs and the composite fermions, which include Yukawa terms
[23] and higher-dimensional terms such as derivative interactions [22]. The
first class of Yukawa terms is responsible for reproduction of the SM Yukawa
couplings after rotating to mass basis and the high-dimensional terms are
instead model-dependent with magnitudes of coefficients in these terms
suppressed by power laws of $f$. As long as $f$ is large enough their
contribution to $a_{\mu}$ is subdominant compared to the Yukawa contribution.
Without losing simplicity, we will neglect these high-dimensional terms. Note
that the Yukawa interactions give rise to mixing effects between the composite
doublets and singlets after electroweak symmetry breaking (EWSB).

  * •

Finally, the mixing terms between the elementary and composite fermions in
$\mathcal{L}_{\rm{m}}$ are essential to explicitly break $\mathcal{G}$ into
$\mathcal{H}$ as characterized by the small mixing angles
$\sin\theta_{\ell_{L}}$ and $\sin\theta_{\ell_{R}}$ etc, which are external to
EWSB. Together with the mixing effects between the composite fermions in
$\mathcal{L}_{s}$ induced by EWSB, these mixing terms allow the composite muon
fermions to couple to both the left- and right-hand muon via $h$
simultaneously. This feature is crucial to understand why the dimension of the
operator in Eq.(1) is fixed to be five in the composite Higgs model, compared
to the dimension of the effective operator either the same as in some
vectorlike models such as [7] or higher in other vectorlike models such as
[13].

Despite the lepton compositeness being small, the Yukawa coupling constant
$g_{*}$ of the composite lepton fermions to the composite Higgs can be as
large as $\sim\sqrt{4\pi}$, which implies that the one-loop composite muon
contribution to $a_{\mu}$ can be significant even with their mass scales
$m_{L}$ and $m_{E}$ bigger than $\upsilon$, as compared to light degree(s) of
freedom required by the vectorlike models, among others.

Along this line both hierarchical and diagonal mixing angles in
$\mathcal{L}_{m}$ in flavor space are technically appropriate to address the
elementary fermion mass hierarchy and small flavor-violation effects
simultaneously. For a quantitative analysis on the composite fermion
contributions to $a_{\mu}$, it is sufficient to make use of a set of
parameters as composed of composite partner masses $m_{L}$ and $m_{E}$, the
strong coupling constant $g_{*}$, and the small mixing angles
$\sin\theta_{\mu_{L}}$ and $\sin\theta_{\mu_{R}}$.

##  3 Muon g-2

In this section we discuss the contributions of new composite fermion partners
to muon $a_{\mu}$. We will show that while the effect of composite quark
partners is negligible, the composite muon partners are able to address the
anomaly.

###  3.1 Composite Quark Partners

Composite quark states contribute to $a_{\mu}$ either via hadron effect in the
low energy region or two-loop electroweak corrections. For the hadron effect,
it goes through photon vacuum polarization [2]

$\displaystyle{}\delta
a^{\rm{had}}_{\mu}\sim\int\frac{R(s)\hat{K}(s)}{s^{2}}ds,$ (5)

where $R(s)$ is the total cross section of electron pair annihilation with $s$
the center-of-mass energy, and $\hat{K}$ is the weight function. As shown in
Eq.(5), the hadron effect is proportional to $s^{-2}$ that dramatically
suppresses the heavy color state contribution. In the composite quark mass
range $m_{\Psi_{C}}>m_{t}$, $\delta a_{\mu}(\rm{had})<1\%\times
a^{SM}_{\mu}(\rm{had})$, where $a^{SM}_{\mu}(\rm{had})\approx(69.0\pm
0.526)\times 10^{-9}$ [2] is the SM hadron contribution.

Given the large coupling $g_{*}$, the two-loop electroweak corrections to
$a_{\mu}$ is dominated by a type of Barr-Zee [25] diagram. In this diagram,
the composite quark partners run in the inner fermion loop which gives an
effective vertex of type $\Gamma_{\gamma\gamma h}$, and the mixing mass
between the elementary and composite muon as a vertex has to be inserted in
the fermion line attached to the external muon in order to flip the muon
chirality. Compared to this diagram, the other two-loop Feynman diagrams as
those of SM [26, 27] are relatively subdominant. Referring to the illustrative
numbers that the model-independent LHC bound on the composite quark mass is
larger than $\sim 1.5$ TeV [28, 29], the model-dependent LHC bound on the
composite muon mass is above $\sim 200$ GeV [30, 31], and the theoretical
limit on the strong coupling $g_{*}\leq\sqrt{4\pi}$, one finds that this
diagram contributes to a value of $\delta a_{\mu}$ beneath the reported value
in Eq.(2) at least by two orders of magnitude. Adjusting the value of $g_{*}$
and/or $m_{\Psi_{\mu}}$ in allowed ranges does not change the situation.

###  3.2 Composite Lepton Partners

The small mixings between the composite muon partners and the elementary muon
in Eq.(3) lead to the following interactions

$\displaystyle{}\mathcal{L}_{\rm{int}}$ $\displaystyle=$
$\displaystyle\left[g^{W}_{NL}\bar{N}\gamma^{\mu}\mu_{L}+g^{W}_{NR}\bar{N}\gamma^{\mu}\mu_{R}\right]W^{+}_{\mu}+h.c$
(6) $\displaystyle+$
$\displaystyle\left[g^{H}_{FL}\bar{\Psi}_{F}\mu_{L}+g^{H}_{FR}\bar{\Psi}_{F}\mu_{R}\right]h+h.c,$

where the couplings are explicitly shown in Table.1. Here, index $F=A,B$ refer
to the two mass eigenstates of the charged left- and right-hand composite muon
partners with mixing angle $\beta$, see Eq.(3.2). In Eq.(6) the absence of
vertex $\Psi_{F}-\mu-\gamma$ is due to the same electric charge
$Q_{\Psi_{\mu}}=Q_{\mu}=-1$, whereas that of vertex $\Psi_{F}-\mu-Z$ follows
from the same weak isospin we have adopted.333 This vertex can be present in
multi-site realizations of the scenario, where the gauge covariant derivative
takes a form $D_{\mu}=\partial_{\mu}-ig^{\prime}YB_{\mu}-ig_{*}\rho_{\mu}$,
with $\rho_{\mu}$ heavy composite gauge bosons to mix with elementary
electroweak gauge bosons. Even so, the effect of this interaction on $a_{\mu}$
is still small compared to that of the Yukawa interaction with large $g_{*}$,
see below Eq.(3.2).

| $A$ | $B$ | $N$
---|---|---|---
$g^{W}_{NL}$ | $-$ | $-$ | $\frac{g}{\sqrt{2}}\sin\theta_{\mu_{L}}$
$g^{W}_{NR}$ | $-$ | $-$ | $0$
$g^{H}_{FL}$ | $-g_{*}\sin\theta_{\mu_{L}}\sin\beta$ | $g_{*}\sin\theta_{\mu_{L}}\cos\beta$ | $-$
$g^{H}_{FR}$ | $g_{*}\sin\theta_{\mu_{R}}\cos\beta$ | $g_{*}\sin\theta_{\mu_{R}}\sin\beta$ | $-$
Table 1: Couplings of the composite lepton partners to the elementary left-
and right-hand muon with index $F=A,B$ in the leading order of
$\sin\theta_{\mu_{L}}$ and/or $\sin\theta_{\mu_{R}}$. See texts for details
about the absences of both $\gamma$\- and $Z$-relevant vertex.

As expected, one obtains the SM muon Yukawa coupling

$\displaystyle{}y_{\mu}\approx g_{*}\sin\theta_{\mu_{L}}\sin\theta_{\mu_{R}},$
(7)

after integrating out the composite muon partners $L$ and $E$. To derive the
masses $m_{F}$ for the eigenstates $\Psi_{F}$ in Eq.(6) it is sufficient to
only consider the mass matrix

$\displaystyle{}\left(\begin{array}[]{cc}\bar{L}_{L}&\bar{E}_{L}\end{array}\right)\left(\begin{array}[]{cc}m_{L}&-g_{*}\upsilon\\\
-g_{*}\upsilon&m_{E}\\\ \end{array}\right)\left(\begin{array}[]{c}L_{R}\\\
E_{R}\\\ \end{array}\right).$ (13)

from $\mathcal{L}_{s}$ and ignore the small mixing effects from
$\mathcal{L}_{m}$ in Eq.(3). Diagonalizing the mass matrix in Eq.(13) yields

$\displaystyle{}m_{A}$ $\displaystyle=$ $\displaystyle
m_{L}\cos^{2}\beta+g_{*}\upsilon\sin 2\beta+m_{E}\sin^{2}\beta,$
$\displaystyle m_{B}$ $\displaystyle=$ $\displaystyle m_{E}\cos^{2}\beta-
g_{*}\upsilon\sin 2\beta+m_{L}\sin^{2}\beta,$ (14)

where the angle $\beta$ satisfies

$\displaystyle{}\left(m_{L}-m_{E}\right)\sin 2\beta=2g_{*}\upsilon\cos
2\beta.$ (15)

Using Table.1 one obtains each contribution to $a_{\mu}$ with respect to the
individual Feynman diagram in Fig.1,

$\displaystyle{}\delta a^{Z}_{\mu}$ $\displaystyle\approx$
$\displaystyle\sum_{F=A,B}-\frac{m^{2}_{\mu}}{16\pi^{2}m^{2}_{Z}}\left[\frac{4}{3}g^{Z}_{FL}g^{Z}_{FR}\frac{m_{F}}{m_{\mu}}\mathcal{C}\left(\frac{m^{2}_{F}}{m^{2}_{Z}}\right)\right]$
$\displaystyle\delta a^{W}_{\mu}$ $\displaystyle\approx$
$\displaystyle-\frac{m^{2}_{\mu}}{16\pi^{2}m^{2}_{W}}\left[-\frac{1}{3}(g^{W}_{NL})^{2}\mathcal{M}\left(\frac{m^{2}_{N}}{m^{2}_{W}}\right)\right]$
$\displaystyle\delta a^{H}_{\mu}$ $\displaystyle\approx$
$\displaystyle\sum_{F=A,B}-\frac{m^{2}_{\mu}}{16\pi^{2}m^{2}_{h}}\left[-\frac{2}{3}g^{H}_{FL}g^{H}_{FR}\frac{m_{F}}{m_{\mu}}\mathcal{F}\left(\frac{m^{2}_{F}}{m^{2}_{h}}\right)\right],$
(16)

in the large composite mass region $m_{F}>m_{h},m_{Z},m_{W}$, where the loop
functions $\mathcal{C}$, $\mathcal{M}$ and $\mathcal{F}$ can be found in
ref.[14].

Figure 1: Feynman diagrams for the new contributions to $a_{\mu}$ due to the
composite states with $\Psi_{F}$ the composite muon states and $N$ the neutral
field.

The correction to $a_{\mu}$ due to the composite lepton partners in Eq.(3.2)
in our model is dominated by the $h$-exchange diagram. Given a fixed $g_{*}$
the value of $\delta a^{H}_{\mu}$ therein, which is proportional to $\sin
2\beta$ as seen in Table.1, maximizes at $\tan\beta=1$ that corresponds to
$m_{L}\approx m_{E}\approx m_{\Psi}$.444In contrast to the degenerate masses,
a large mass splitting between $m_{L}$ and $m_{E}$ leads to $\tan\beta\sim 0$
which suppresses the magnitude of $\delta a^{H}_{\mu}$. Under this choice
$\delta a^{H}_{\mu}$ explicitly reads

$\displaystyle{}\delta
a^{H}_{\mu}\approx\frac{g_{*}y_{\mu}}{2\pi^{2}}\frac{m_{\mu}}{m_{\Psi}}f\left(\frac{g_{*}\upsilon}{m_{\Psi}}\right),$
(17)

with the function

$\displaystyle{}f(x)=\frac{x}{(1-x^{2})[(1-x)^{-2}+(1+x)^{-2}]},$ (18)

where the small mixing angles $\sin\theta_{\mu_{L,R}}$ have been replaced by
$y_{\mu}$ in Eq.(7), and the rational region of $x$ corresponds to $0<x<1$.
Eq.(17) shows that $\delta a^{H}_{\mu}$ are sensitive to $g_{*}$, $m_{\Psi}$
and $x$. To resolve the muon anomaly, a large coupling $g_{*}$ seems
necessary. For instance, with $g_{*}\sim 3$ and $m_{\Psi}\sim 1$ TeV the
reported muon g-2 anomaly is easily resolved. We will present more details
about the viable parameter space based on Eq.(17) in Sec.5 after we
investigate the constraints in the next section.

##  4 Constraints

###  4.1 Indirect Constraints

####  4.1.1 Higgs Portal

The strong coupling of $\Psi_{F}$ to the composite Higgs is the key ingredient
to address the muon anomaly, which can be constrained by precise measurements
on the Higgs couplings. For the couplings of Higgs to SM gauge bosons we
consider the one-loop modification to the coupling of $h$ to di-photon as in
the effective operator $c_{\gamma}hF^{2}_{\mu\nu}$ due to the composite muon
partners. Since this operator explicitly breaks the global symmetry of $h$,
$c_{\gamma}$ is therefore controlled by the mixing terms in
$\mathcal{L}_{\rm{m}}$ in Eq.(3). Inserting them into the one-loop fermion
triangle diagram as vertexes gives rise to [32]

$\displaystyle{}\delta
c_{\gamma}\approx\frac{\alpha}{2\pi}\frac{g^{2}_{*}\upsilon}{m^{2}_{\Psi}}\sin^{2}\theta_{\mu_{R}}$
(19)

in the mass region $m_{h}<m_{\Psi}<m_{*}\approx g_{*}f$. The dependence of
$\delta c_{\gamma}$ on the $\sin\theta_{\mu_{R}}$ in Eq.(19) implies that the
effect of the composite muon fermions on the decay $h\rightarrow\gamma\gamma$
is greatly suppressed as a result of the small $\mu$ compositeness so that it
is easily beyond current LHC precision [33, 34]. Just like
$h\rightarrow\gamma\gamma$ the decay mode $h\rightarrow Z\gamma$ [35] imposes
similar constraint, even though the later one is not always independently
analyzed by LHC experiments.

For the coupling of Higgs to SM fermions, the precision tests on the SM Yukawa
couplings provide information on the composite mass scale as

$\displaystyle{}\frac{\delta
y_{\psi}}{y_{\psi}}\sim\frac{\upsilon^{2}}{f^{2}}.$ (20)

Taking a conservative lower bound $f\geq 1$ TeV [36] inferred from current LHC
data implies that the composite mass scale $m_{*}\geq g_{*}$ TeV.

####  4.1.2 $Z$ Portal

The mixing effects between the elementary and the composite muon field modify
the $Z-\mu-\mu$ couplings as

$\displaystyle{}g^{Z}_{\mu_{L}\mu_{L}}$ $\displaystyle\approx$
$\displaystyle\left(-\frac{1}{2}+s^{2}_{W}\right)(1+\sin^{2}\theta_{\mu_{L}})$
$\displaystyle g^{Z}_{\mu_{R}\mu_{R}}$ $\displaystyle\approx$ $\displaystyle
s^{2}_{W}(1+\sin^{2}\theta_{\mu_{R}}),$ (21)

where unity in the bracket means the SM values with $s_{W}$ the weak mixing
angle. Imposing the LEP constraint $\delta g^{Z}_{\mu_{L}\mu_{L}}\leq 10^{-3}$
[37] implies $\sin\theta_{\mu_{L}}\leq 3.3\times 10^{-2}$, which is far above
the lower bound $\sin\theta_{\mu_{L}}\geq 10^{-4}$ as seen from the SM muon
Yukawa coupling in Eq.(7). Note that the corrections in Eq.(4.1.2) are model-
dependent as they depend on the isospins of the composite muon partners.

####  4.1.3 Oblique Corrections

The last precise tests are oblique parameters [38]. The oblique parameter
$\hat{S}$ tied to $W^{3}_{\mu}$ and $B_{\mu}$ gauge boson receives corrections
from the composite vector bosons $\rho_{\mu}$ at tree level and the composite
Higgs and fermions at loop level, respectively. The tree-level correction
$\hat{S}(\rho_{\mu})\sim m^{2}_{W}/m^{2}_{*}$ implies that $m_{*}\geq 3$ TeV.
Moreover, the oblique parameter $\hat{T}$ does not receive tree-level
correction due to the protection of custodial symmetry. $\hat{T}$ only obtains
loop corrections due to the composite Higgs and fermions. Similar to
vectorlike doublet $+$ singlet fermion contributions [40, 39], the one-loop
corrections to $\hat{S}$ and $\hat{T}$ due to the composite muon partner loops
are

$\displaystyle{}\Delta\hat{S}$ $\displaystyle\sim$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\xi\ln\left(\frac{m_{*}}{m_{\Psi}}\right),$
$\displaystyle\Delta\hat{T}$ $\displaystyle\sim$
$\displaystyle\frac{1}{16\pi^{2}}\xi\sin^{4}\theta_{\mu_{L}}\frac{m_{\Psi}^{2}}{f^{2}},$
(22)

with $\xi=\upsilon^{2}/f^{2}$. The correction to the $\hat{T}$ parameter in
Eq.(4.1.3) is greatly suppressed by the small factor
$\sin^{4}\theta_{\mu_{L}}$ as upper bounded by the $Z$ coupling in Eq.(4.1.2).
Likewise, given $\xi\leq 0.1$ inferred from Eq.(20) the correction to the
$\hat{S}$ parameter in Eq.(4.1.3) can be safely neglected as well.

###  4.2 Direct Constraints

Let us now consider direct detections on the parameter space with respect to
the muon anomaly. The couplings and the small mixings of the composite muon
partners $\Psi_{F}$ are the key factors to develop strategies of searching
them at colliders, which are obviously different from studies on composite
quark partners [41, 42, 43], composite gauge bosons [44, 45, 46], and
composite Higgs [47]. After produced at colliders, the composite muon partners
decay through the modes $\Psi^{\pm}_{F}\rightarrow W^{\pm}\nu_{\mu}$,
$\Psi^{\pm}_{F}\rightarrow h\mu^{\pm}$ and $\Psi^{\pm}_{F}\rightarrow
W^{\pm}N$ for the heavier state $F=A$. For a practical purpose, it is
sufficient to focus on the signals of the lighter state $\Psi_{B}$ at the
14-TeV LHC.

For simulation of signals and relevant SM backgrounds in the following, we use
FeynRules [48] to generate model files prepared for MadGraph5 [49] that
includes Pythia 8 [50] for parton showering and hadronization and the package
Delphes 3 [51] for fast detector simulation.

Figure 2: Feynman diagram for the Drell-Yan productions of $\Psi_{B}$ pair at
the LHC with the decay mode $\Psi^{\pm}_{B}\rightarrow h\mu^{\pm}$.

####  4.2.1 Gauge Boson Portal

The $\Psi_{B}$ pair can be produced either through $e^{+}e^{-}\rightarrow
Z^{*}/\gamma^{*}\rightarrow\Psi^{+}_{B}\Psi^{-}_{B}$ at LEP or Drell-Yan
processes $pp\rightarrow Z^{*}/\gamma^{*}\rightarrow\Psi^{+}_{B}\Psi^{-}_{B}$
at LHC. Here, do not confuse the coupling $\bar{\Psi}_{F}-\Psi_{F}-Z$ with the
coupling $\bar{\Psi}_{F}-\mu-Z$. So far, the LEP-2 has excluded the mass range
$m_{B}\leq 101.2$ GeV [52], while the LHC has reported a limit $m_{B}\geq 200$
GeV based on the decay mode $\Psi^{\pm}_{B}\rightarrow Z\mu^{\pm}$ [31] in
certain vectorlike lepton models. We can revisit this LHC bound by repeating
the Drell-Yan processes with Feynman diagram shown in Fig.2, where the decay
mode $\Psi^{\pm}_{B}\rightarrow Z\mu^{\pm}$ is replaced by our decay mode
$\Psi^{\pm}_{B}\rightarrow h\mu^{\pm}$. Followed by the decay of
$h\rightarrow\bar{b}b$, the primary SM backgrounds corresponding to this
signal is given by $pp\rightarrow Z/\gamma(\ell^{+}\ell^{-})+{\rm jets}$.

Fig.3 shows the cross sections of the Drell-Yan processes for the lighter
composite muon state $\Psi_{B}$ at the 14-TeV LHC with $g_{*}=\sqrt{4\pi}$
that represents a typical value of $g_{*}$ in the parameter space obtained
from the muon g-2 anomaly. In this figure, the virtual $Z$ contribution is
subdominant in comparison with the virtual photon contribution. This follows
from the fact that the coupling of $\Psi_{B}$ pair to $Z$ is relatively
smaller than the coupling of $\Psi_{B}$ pair to $\gamma$ as in the SM. The SM
background for this signal comes from both $pp\rightarrow
Z(\ell^{+}\ell^{-})+{\rm jets}$ and
$pp\rightarrow\gamma(\ell^{+}\ell^{-})+{\rm jets}$, with leading-order cross
section $4.8\times 10^{6}$ fb and $6.5\times 10^{6}$ fb respectively. Combing
them gives a total cross section about $1.13\times 10^{7}$ fb, implying that
the signal cross section as shown in Fig.3 is at least six order smaller than
its SM background cross section.

Figure 3: Cross sections of the Drell-Yan processes in Fig.2 as function of
the lighter mass $m_{B}$ at the 14-TeV LHC with $g_{*}=\sqrt{4\pi}$, as
compared to the total cross section of the SM background about $1.13\times
10^{7}$ fb.

In order to distinguish the above signal from its SM background, we apply the
latest CMS cuts [53, 54] on four-b jets and the CMS cuts [55] on two muons
with same flavor but opposite sign

$\displaystyle{}{p_{T}}_{j_{1(2,3,4)}}$ $\displaystyle>$ $\displaystyle
45~{}{\rm GeV},$ $\displaystyle{p_{T}}_{\mu_{1(2)}}$ $\displaystyle>$
$\displaystyle 20~{}{\rm GeV},$ $\displaystyle|\eta_{j_{1(2,3,4)}}|$
$\displaystyle<$ $\displaystyle 2.5,$ $\displaystyle|\eta_{\mu_{1(2)}}|$
$\displaystyle<$ $\displaystyle 2.4,$ (23)

to our Drell-Yan processes, where ${p_{T}}_{j_{1(2,3,4)}}$ and
$\eta_{j_{1(2,3,4)}}$ are the transverse momentum and pseudo-rapidity of jet
respectively, while ${p_{T}}_{\mu_{1(2)}}$ and $\eta_{\mu_{1(2)}}$ are the
transverse momentum and pseudo-rapidity of muon respectively. Besides the cuts
in Eq.(4.2.1), any event with an additional jet with $p_{T}>30$ GeV or muon
with $p_{T}>10$ GeV is rejected. The cuts in Eq.(4.2.1) can dramatically
reduce the number of SM events arising from
$pp\rightarrow\gamma(\ell^{+}\ell^{-})+{\rm jets}$, which makes the SM
background of the Drell-Yan processes similar to that of the following
$h$-associated processes.

Figure 4: Feynman diagrams for the $h$-associated single (left) and pair
(right) production of $\Psi_{B}$ at the LHC.

####  4.2.2 Higgs Portal

Through Higgs $\Psi_{B}$ can be singly produced by $pp\rightarrow
Zh^{*}\rightarrow Z\Psi^{\pm}_{B}\mu^{\mp}$ which is suppressed by the small
mixing angles $\theta_{\mu_{L}}$ and $\theta_{\mu_{R}}$, or produced in pair
by $pp\rightarrow\Psi^{+}_{B}\Psi^{-}_{B}jj^{\prime}$ which is instead not
suppressed by the small mixing angles. Note that the jets $j$ and $j^{\prime}$
can be either the same or different. Fig.4 shows the Feynman diagrams for
these two processes. In Fig.4(a), the single production $pp\rightarrow
Zh^{*}\rightarrow l^{+}l^{-}\Psi^{\pm}_{B}\mu^{\mp}\rightarrow
l^{+}l^{-}h\mu^{+}\mu^{-}$ is followed by the SM decay $Z\rightarrow
l^{+}l^{-}$ with $l=e,\mu$ and the decay mode $\Psi^{\pm}_{B}\rightarrow
h\mu^{\pm}$, for which the SM background is $pp\rightarrow$ mono-Z+jets with
cross section of $1.9\times 10^{2}$ fb. In Fig.4(b), the pair production
$pp\rightarrow\Psi^{+}_{B}\Psi^{-}_{B}jj^{\prime}\rightarrow
hh\mu^{+}\mu^{-}jj^{\prime}$ is followed by the decay mode
$\Psi^{\pm}_{B}\rightarrow h\mu^{\pm}$, for which the SM background is
$pp\rightarrow Z+$ jets with cross section of $4.8\times 10^{6}$ fb.

Fig.5 shows the cross sections of the two processes in Fig.4 at the 14-TeV LHC
with $g_{*}=\sqrt{4\pi}$. In the case of single production we have
additionally chosen the equal small mixing angles
$\theta_{\mu_{L}}\approx\theta_{\mu_{R}}$. Adjusting the allowed values of
these small mixing angles may yield a larger cross section for the single
production. However, it is still not enough to let this process be important.
In contrast, the pair production seems more promising, as the gap between the
cross sections of the signal and SM background is smaller in the mass range
$m_{B}$ less than 300 GeV. As the final states of this signal contain four
b-jets, the CMS cuts in the previous Drell-Yan processes can be similarly
applied to it. Then, one recognizes that the SM background of the
$h$-associated pair production is similar to that of the Drell-Yan processes.
Given the same $m_{B}$ and $g_{*}$, the signal cross section in the
$h$-associated pair production in Fig.5 is smaller than in the Drell-Yan
productions in Fig.3 by roughly an order of magnitude, leading to the later
one more competitive than the former one. We will neglect the $h$-associated
pair production in the following numerical analysis.

Figure 5: Cross section of the single (in blue) and pair (in red) of
$\Psi_{B}$ in Fig.4 as function of $m_{B}$ at the 14-TeV LHC with
$g_{*}=\sqrt{4\pi}$. Compared to the pair production, the cross section of
single production is suppressed by the small mixing angles.

##  5 Numerical Results

Collecting the parameter space in Sec.3 and the indirect and direct
constraints in Sec.4, we present the main numerical results in Fig.6 for
$\sin\theta_{\mu_{L}}=\sin\theta_{\mu_{R}}$. In this figure, the parameter
space (in green band) means the solution to the muon g-2 anomaly within a
discrepancy of $2\sigma$, where $g_{*}\sim 1-3$ and $m_{B}\sim 200-550$ GeV.
Unlike light degree(s) of freedom in vectorlike models as required to resolve
the anomaly, the allowed large mass range of $m_{B}$ manifests the composite
Higgs model. In the light of the direct limits from the Drell-Yan productions
of the lighter composite state $\Psi_{B}$ at the 14-TeV LHC with the
integrated luminosity of 3 ab-1 with 2$\sigma$ exclusion (in red curve) and
5$\sigma$ discovery (in blue curve)555We follow the criteria $S/\sqrt{B}=1.96$
for exclusion and $S/\sqrt{S+B}=5$ for discovery, where $S$ and $B$ refer to
the event number of signal and SM background, respectively., the parameter
regions with $m_{B}$ up to at most $\sim 255$ GeV can be discovered and up to
$\sim 325$ GeV excluded, respectively.

Figure 6: Parameter space (in green) with respect to the muon g-2 anomaly for
$\sin\theta_{\mu_{L}}=\sin\theta_{\mu_{R}}$, and the prospect of searching for
the parameter regions based on the Drell-Yan processes of $\Psi_{B}$ at the
14-TeV LHC with an integrated luminosity 3 ab-1, with 2$\sigma$ exclusion (in
red) and 5$\sigma$ discovery (in blue) shown simultaneously. See text for
details.

A few comments are in order regarding the numerical results in Fig.6.

  * •

With $\sin\theta_{\mu_{L}}=\sin\theta_{\mu_{R}}$ all of the indirect
constraints in Sec.4.1 can be satisfied by $f$ larger than $\sim 3/g_{*}$ TeV.
In particular, the deviation of the decay width of $h\rightarrow\gamma\gamma$
relative to the SM reference value due to the composite muon partners is of
order $\sim 10^{-8}\sim 10^{-9}$ in the parameter regions of Fig.6. In
practice, this indirect constraint is more viable for the composite top
partners depending on the top partner mass scale. As the magnitude of the
compositeness parameters grows from $\sim 0.01$ to $\sim 0.1-1$ as seen in
Eq.(7), the effect of the top partners on this decay is significantly
amplified.

  * •

The direct limits arising from the Drell-Yan productions are more sensitive to
$m_{B}$ than $g_{*}$. The reason is that the signal cross section is affected
by $g_{*}$ only through the dependence of the branching ratio of the decay
mode $\Psi^{\pm}_{B}\rightarrow h\mu^{\pm}$ on this parameter, which is nearly
fixed in the parameter regions with $m_{B}\geq 200$ GeV and $g_{*}\geq 1$. We
have verified this point by observing that the cross section in Fig.3 hardly
change as $g_{*}$ varies.

  * •

As noticed in Eq.(3.2), the parameter space is rather sensitive to the mixing
effects between the composite muon states. If one let the input masses $m_{L}$
and $m_{E}$ be highly non-degenerate, the parameter space will be obviously
reduced or even disappear.

##  6 Conclusion

In this work we have considered the scenario of composite Higgs with partial
compositeness as the solution to the muon anomalous magnetic moment as
recently confirmed by the FNAL experiment. The anomaly is resolved by the one-
loop correction to $a_{\mu}$ due to the composite muon partners with their
mass scale of order TeVs and strong coupling constant $g_{*}$ of order unity.
Compared with alternative scenarios where light states are required, the
relatively large mass regions can manifest the composite Higgs model. We have
presented the parameter space by taking into account the constraints of
precise tests on $h$, $Z$ and oblique electroweak parameters, and explored the
prospect of searching for the parameter regions in terms of both the Drell-Yan
and $h$-associated productions of the lighter composite muon partner at the
14-TeV LHC with the integrated luminosity 3 ab-1. To draw the direct detection
limits, we have applied the latest CMS cuts about four b-jets to both the
Drell-Yan and the $h$-associated pair productions. It turns out that the
parameter regions with the lighter composite muon mass below $\sim 325$ GeV
can be excluded at $2\sigma$ order in the Drell-Yan productions, despite the
ability of exclusion in the $h$-associated productions being weak. Finally,
although not discussed here, this scenario may be linked to other leptonic
anomalies.

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China (No. 11775039) and the High-level Talents Research and Startup
Foundation Projects for Doctors of Zhoukou Normal University
(No.ZKNUC2021006).

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#  GARField: Group Anything with Radiance Fields

Chung Min Kim∗1 Mingxuan Wu∗1 Justin Kerr∗1 Ken Goldberg1
Matthew Tancik2 Angjoo Kanazawa1
$*$ Denotes equal contribution
1UC Berkeley 2 Luma AI

###### Abstract

Grouping is inherently ambiguous due to the multiple levels of granularity in
which one can decompose a scene—should the wheels of an excavator be
considered separate or part of the whole? We present Group Anything with
Radiance Fields (GARField), an approach for decomposing 3D scenes into a
hierarchy of semantically meaningful groups from posed image inputs. To do
this we embrace group ambiguity through physical scale: by optimizing a scale-
conditioned 3D affinity feature field, a point in the world can belong to
different groups of different sizes. We optimize this field from a set of 2D
masks provided by Segment Anything (SAM) in a way that respects coarse-to-fine
hierarchy, using scale to consistently fuse conflicting masks from different
viewpoints. From this field we can derive a hierarchy of possible groupings
via automatic tree construction or user interaction. We evaluate GARField on a
variety of in-the-wild scenes and find it effectively extracts groups at many
levels: clusters of objects, objects, and various subparts. GARField
inherently represents multi-view consistent groupings and produces higher
fidelity groups than the input SAM masks. GARField’s hierarchical grouping
could have exciting downstream applications such as 3D asset extraction or
dynamic scene understanding. See the project website at
https://www.garfield.studio/

Figure 1: Group Anything with Radiance Fields (GARField) We present GARField,
which distills multi-level groups represented as masks into NeRF to create a
scale-conditioned 3D affinity field (top left). Once trained, this affinity
field can be clustered at a variety of scales to decompose the scene at
different levels of granularity, like breaking apart the excavator into its
subparts (bottom). 3D assets can be extracted from this hierarchy by
extracting every group in the scene automatically or via user clicks, as
visualized here (top right).

##  1 Introduction

Consider the scene in Figure 1. Though recent technologies like NeRFs [20] can
recover photorealistic 3D reconstructions of this scene, the world is modeled
as a single volume with no structural meaning. As humans, not only can we
reconstruct the scene, but we also have the ability to group it at multiple
levels of granularity — at the highest level, we see the parts of the scene
_i.e_. the excavator, bushes, and the sidewalk, but we are also able to
decompose the excavator into its parts such as its wheels, crane, and the
cabin. This ability to perceive the scene at multiple levels of groupings is a
key component of our scene understanding, enabling us to interact with the 3D
world by understanding what belongs together. However, these different levels
of granularity introduce ambiguity in groups, making it a challenge to
represent them in a coherent 3D representation. While there are multiple ways
to break this ambiguity, we focus on the physical scale of entities as a cue
to consolidate groups into a hierarchy.

In this work we introduce Group Anything with Radiance Fields (GARField), an
approach that, given posed images, reconstructs a 3D scene along with a scale-
conditioned affinity field that enables decomposing the scene into a hierarchy
of groups. For example, GARField can extract both the entire excavator (Fig. 1
Top Right) as well as its subparts (Bottom Right). This dense hierarchical 3D
grouping enables applications such as 3D asset extraction and interactive
segmentation.

GARField distills a set of 2D segmentation masks into a 3D volumetric scale-
conditioned affinity field. Because grouping is an ambiguous task, these 2D
labels can be overlapping or conflicting. These inconsistencies pose a
challenge for distilling masks into consistent 3D groups. We overcome this
issue by leveraging a scale-conditioned feature field. Specifically GARField
optimizes a dense 3D feature field which is supervised such that feature
distance reflects points’ affinity. The scale conditioning enables two points
to have higher affinity at a large scale but low affinity at a smaller scale
(_i.e_. wedges of the same watermelon), as illustrated in Figure 2.

Though in principle GARField can distill any source of 2D masks, we derive
mask candidates from Segment Anything Model (SAM) [15] because they align well
with what humans consider as reasonable groups. We process input images with
SAM to obtain a set of candidate segmentation masks. For each mask, we compute
a physical scale based on the scene geometry. To train GARField, we distill
candidate 2D masks with a contrastive loss based on mask membership,
leveraging 3D scale to resolve inconsistencies between views or mask
candidates.

A well-behaved affinity field has: 1) transitivity , which means if two points
are mutually grouped with a third, they should themselves be grouped together,
and 2) containment, which means if two points are grouped at a small scale,
they should be grouped together at higher scales. GARField’s use of
contrastive loss in addition to a containment auxiliary loss encourages both
of these properties.

With the optimized scale-conditioned affinity field, GARField extracts a 3D
scene hierarchy via recursively clustering them at descending scales until no
more clusters emerge. By construction, this recursive clustering ensures that
generated groups are subparts of the prior cluster in a coarse-to-fine manner.
We evaluate GARField on a variety of real scenes with annotated hierarchical
groupings, testing its ability to capture object hierarchy, and its
consistency across different views. By leveraging multiple views, GARField is
able to produce detailed groupings, often improving upon the quality of input
2D segmentation masks. Moreover, these groups are 3D consistent by design,
while 2D baselines do not guarantee view consistency. We show downstream
applications of GARField for hierarchical 3D asset extraction and click-based
interactive segmentation. Given GARField’s scene decomposition capabilities,
we’re hopeful for its potential in other downstream applications like enabling
robots to understand they can interact with or as a prior for dynamic
reconstruction. Code and data will be released upon publication. Please see
the supplemental video for more visualizations.

Figure 2: Importance of Scale When Grouping A single point may belong to
multiple groups. GARField uses scale-conditioning to reconcile these
conflicting signals into one affinity field.

##  2 Related Work

Figure 3: GARField Method: (Left) given an input image set, we extract a set
of candidate groups by densely querying SAM, and assign each a physical scale
by deprojecting depth from the NeRF. These scales are used to train a scale-
conditioned affinity field (Right). During training, pairs of sampled rays are
pushed apart if they reside in different masks, and pulled together if they
land in the same mask. Affinity is supervised only at the scale of each mask,
which helps resolve conflicts between them.

##### Hierarchical Grouping

Multi-level grouping has long been studied in 2D images since the early days
of foreground segmentation [28]. Several methods build on this idea of
spectral clustering for multi-level segmentation [5] and more complex
hierarchical scene parsing [1, 25, 31]. These approaches rely on extracting
contours either via classic texture cues and create a hierarchy either via a
top-down [37] or bottom-up consolidation [1]. More recent deep learning
approaches use edges [36] computed at multiple scales to create the hierarchy,
and Ke _et al_. [11] proposes a transformer based unsupervised hierarchical
segmentation approach guided by the outputs of a classic hierarchical
segmentation [1].

Many works circumvent the question of ambiguity in grouping by defining a set
of categories within which instances are to be segmented, _i.e_. panoptic
segmentation [10, 14]. Recently, Segment Anything (SAM) [15] off-loads this
ambiguity into prompting, where at each pixel multiple segmentation masks can
be proposed. However SAM does not recover a consistent set of hierarchical
groups in the scene, which we enable by multi-scale 3D distillation.

Hierarchical part decomposition has also been explored in 3D objects, either
in a supervised [35, 17, 21], or unsupervised manner [24]. Our approach
distills information from a 2D model, and we consider full scenes while these
approaches focus on 3D objects.

Segmentation in NeRFs Existing approaches for segmentation in NeRFs typically
distill segmentation masks into 3D either by using ground-truth semantic
labels [38, 29], matching instance masks [18], or training 3D segmentation
networks on NeRF [34]. However, these techniques do not consider hierarchical
grouping, and are only interested in a flat hierarchy of objects or instances.
Ren et al. [27] leverages human interaction in the form of image scribbles to
segment objects with interaction. More recently, Cen et al. [3] try to recover
a 3D consistent mask from SAM by tracking the 2D masks between neighboring
views via user prompting. Chen et al. [4] attempt this by distilling SAM
encoder features into 3D and querying the decoder. In contrast with these
approaches, our approach GARField does not require user input; it is able to
obtain a hierarchical grouping of the scene automatically, and furthermore the
recovered groups are view-consistent by definition.

3D Feature Fields Distilling higher-dimensional features into a neural field,
in tandem with a radiance field (view-dependent color and density), has been
thoroughly explored. Methods like Semantic NeRF [38], Distilled Feature Fields
[16], Neural Feature Fusion Fields [33], and Panoptic Lifting [29] distill
per-pixel 2D features into 3D by optimizing a 3D feature field to reconstruct
the 2D features after volumetric rendering. These features can be either from
pretrained vision models such as DINO or from semantic segmentation models.
LERF [13] extends this idea to a scale-conditioned feature field, enabling the
training of feature fields from global image embeddings like CLIP [26].
GARField similarly optimizes a scale-conditioned feature field in 3D; however,
the purpose of the multi-scale features is to resolve ambiguity in grouping,
instead of reconstructing an explicit 2D feature like CLIP. In addition LERF
has no spatial grouping, a shortcoming GARField addresses. The aforementioned
methods are based on direct supervision from image features, while other
methods such as NeRF-SOS [8] and Contrastive Lift [2] optimize an arbitrary
feature field at a single scale using a contrastive loss between pairs of rays
based on similarity. GARField uses this contrastive approach because it allows
for defining pairwise relationships between points based on mask labels.
However, we design a scale-conditioned contrastive loss, which allows for
distilling conflicting masks into 3D. In addition, GARField does not require
the slow-fast formulation of Bhalgat et al. [2] for stable training, perhaps
enabled by scale-conditioned training.

##  3 Method

###  3.1 2D Mask Generation

GARField takes as input a set of posed images and produces a hierarchical 3D
grouping of the scene, along with a standard 3D volumetric radiance field and
a scale-conditioned affinity field. To do this, we first pre-process input
images with SAM to obtain mask candidates. Next, we optimize a volumetric
radiance field along with the affinity field which takes in a single 3D
location and a euclidean scale, and outputs a feature vector. Affinity is
obtained by comparing pairs of points’ feature vectors. After optimization,
the resulting affinity field can be used to decompose a scene by recursively
clustering the feature embeddings in 3D at descending scales in a coarse-to-
fine manner, or for segmenting user specified queries. The overall pipeline is
illustrated in Figure 3.

In order to train a GARField, we first mine 2D mask candidates from an image
and then assign a 3D scale for each mask. Specifically, we use SAM’s automatic
mask generator [15], which queries SAM in a grid of points and produces 3
candidate segmentation masks per query point. Then, it filters these masks by
confidence and deduplicates nearly identical masks to produce a list of mask
candidates of multiple sizes which can overlap or include each other. This
process is done independently of viewpoint, producing masks which may not be
consistent across views. In this work we aim to generate a hierarchy of
groupings based on objects’ physical size. As such, we assign each 2D mask a
physical 3D scale as in Fig. 3. To do this we partially train a radiance field
and render a depth image from each training camera pose. Next, for each mask
we consider the 3D points within that mask and pick the scale based on the
extent of the points’ position distribution. This method ensures the 3D scale
of masks resides in the same world-space, enabling scale-conditioned affinity.

Figure 4: Densified Scale Supervision: Consider two grapes within a cluster.
Naively using scale for contrastive loss supervises affinities only at the
grape and grape trio levels, leaving entire intervals unsupervised. In
GARField, we densify the supervision by 1) augmenting scale between mask
euclidean scales and 2) imposing an auxiliary loss on containment of larger
scales.

###  3.2 Scale-Conditioned Affinity Field

Scale-conditioning is a key component of GARField which allows consolidating
inconsistent 2D mask candidates: The same point may be grouped in several ways
depending on the granularity of the groupings desired. Scale-conditioning
alleviates this inconsistency because it resolves ambiguity over which group a
query should belong to. Under scale-conditioning, conflicting masks of the
same point no longer fight each other during training, but rather can coexist
in the same scene at different affinity scales.

We define the scale-conditioned affinity field $F_{\text{g}}(x,s)\mapsto
R^{d}$ over a 3D point $x$ and euclidean scale $s$, similar to the formulation
in LERF [13]. Output features are constrained to a unit hyper-sphere, and the
affinity between two points at a scale is defined by
$A(x_{1},x_{2},s)=-||F_{\text{g}}(x_{1},s)-F_{\text{g}}(x_{2},s)||_{2}$. These
features can be volumetrically rendered with a weighted average using the same
rendering weights based on NeRF density to obtain a value on a per-ray basis.

Figure 5:  3D Asset Extraction with Interactive Selection: Users can
interactively select view-consistent 3D groups with GARField using a click
point and a scale.  Figure 6: 3D Decomposition: GARField can be recursively
queried at decreasing scale to cluster a scene into objects and their
subparts. Figure 7: Results: From a GARField we extract objects from the
global scene by selecting top-level clusters, then visualize their local
clusters at decreasing scales. GARField can produce complete 3D object masks,
and break these objects into meaningful subparts based on the input masks. We
use Gaussian Splats [12] to produce these visualizations in 3D. See the
Supplemental video for more results.

####  3.2.1 Contrastive Supervision

The field is supervised with a margin-based contrastive objective, following
the definition provided by DrLIM [9]. There are two core components of the
loss: at a given scale, one which pulls features within the same group to be
close, and another which pushes features in different groups apart.

Specifically, consider two rays $r_{A},r_{B}$ sampled from masks
$\mathcal{M}_{A},\mathcal{M}_{B}$ within the same training image, with
corresponding scales $s_{A}$ and $s_{B}$. We can volumetrically render the
scale-conditioned affinity features along each ray to obtain ray-level
features $F_{A}$ and $F_{B}$. If $\mathcal{M}_{A}=\mathcal{M}_{B}$, the
features are pulled together with L2 distance:
$\mathcal{L}_{\text{pull}}=||F_{A}-F_{B}||$. If
$\mathcal{M}_{A}\neq\mathcal{M}_{B}$, the features are pushed apart:
$\mathcal{L}_{\text{push}}=\text{ReLU}(m-||F_{A}-F_{B}||)$ where $m$ is the
lower bound distance, or margin. Importantly, this loss is only applied among
rays sampled from the same image, since masks across different viewpoints have
no correspondence.

####  3.2.2 Densifying Scale Supervision

The supervision provided by the previous contrastive losses alone are not
sufficient to preserve hierarchy. For example in Fig. 4, although the egg is
correctly grouped with the soup at scale 0.22, at a larger scale it fragments
apart. We hypothesize this grouping instability is because 1) scale
supervision is defined sparsely only when a mask exists and 2) nothing imposes
containment such that small scale groups remain at larger scales. We address
these shortcomings here by introducing the following modifications:

Continuous scale supervision By using 3D mask scales, groups are only defined
at discrete values where masks are chosen. This results in large unsupervised
regions of scale, as shown at the top of Fig. 9. We densify scale supervision
by augmenting the scale $s$ uniformly randomly between the current mask’s
scale and the next smallest mask’s scale. When a ray’s mask is the smallest
mask for the given viewpoint, we interpolate between 0 and $s_{0}$. This
ensures continuous scale supervision throughout the field leaving no
unsupervised regions.

Containment Auxiliary Loss: If two rays $r_{1}$ and $r_{2}$ are in the same
mask with scale $s$, then they should also be pulled together at any scale
larger than $s$. Intuitively, two grapes within the same cluster (Fig. 4) are
also grouped together at larger scales (e.g., the entire bunch). At each
training step, for the rays grouped together at scale $s$, we additionally
sample a larger scale $s^{\prime}>s$ at which the rays are also pulled
together. This ensures that affinities at smaller scales are not lost at
larger scales.

####  3.2.3 Ray and Mask Sampling

Just like standard NeRF training, we sample rays over which to compute losses.
Because GARField uses a contrastive loss within each train image, naively
sampling pixels uniformly during training is inadequate to provide a training
signal in each minibatch of rays. To ensure sufficient pairs in each train
batch, we first sample N images, and sample M rays within each image. To
balance the number of images as well as the number of point pairs for
supervision, we sample 16 images and 256 points per image, resulting in 4096
samples per train iteration.

For each ray sampled, we must also choose a mask to use as the group label for
the train step in question. To do this, we retain a mapping from pixels to
mask labels throughout training, and at each train step randomly select a mask
for each ray from its corresponding list of masks. There are two important
caveats in this sampling process: 1) The probability a mask is chosen is
weighted inversely with the log of the mask’s 2D pixel area. This prevents
large scales from dominating the sampling process, since larger masks can be
chosen via more pixels. 2) During mask selection we coordinate the random
scale chosen across rays in the same image to increase the probability of
positive pairs. To do this, we sample a single value between 0 and 1 per
image, and index into each pixel’s mask probability CDF with the same value,
ensuring pixels which land within the same group are assigned the same mask.
Otherwise, the loss is dominated by pushing forces which destabilize training.

###  3.3 Implementation Details

The method is built in Nerfstudio [32] on top of the Nerfacto model by
defining a separate output head for the grouping field. The grouping field is
represented with a hashgrid [23] with 24 layers and a feature dimension of 2
per layer, and a 4-layer MLP with 256 neurons and ReLU activation which takes
in scale as an extra input concatenated with hashgrid feature. We cap scale at
$2\times$ the extent of cameras, and normalize the scale input to the MLP
using sklearn’s quantile transform on the distribution of computed 3D mask
scales (Sec 3.1). Output embeddings are $d=256$ dimensions. Gradients from the
affinity features do not affect the RGB outputs from NeRF, as these
representations share no weights or gradients.

We begin training the grouping field after 2000 steps of NeRF optimization,
giving geometry time to converge. In addition, to speed training we first
volumetrically render the hash value, then use it as input to the MLP to
obtain a ray feature. With this deferred rendering, the same ray can be
queried at different scales with only one extra MLP call. We normalize the
result of volume rendering to unit norm before inputting to the MLP, and for
point-wise queries, the individual hashgrid value is normalized. Preprocessing
SAM masks takes around 3-10 minutes, followed by about 20 minutes for training
on a GTX 4090.

##  4 Hierarchical Decomposition

Once we have optimized a scale-conditioned affinity, GARField generates a
hierarchy of 3D groups, organized in a tree such that each node is broken into
potential sub-groups. To do this we recursively cluster groups by decreasing
the scale for affinity, using HDBSCAN [19], a density based clustering
algorithm which does not require a prior on number of clusters. This
clustering process can be done in 2D on volumetrically rendered features in an
image which yields masks, or in 3D across points to yield pointclouds. See
Fig. 6 for a visualization of scene decomposition.

Initialization: First, to initialize the hierarchy, we first globally cluster
features at a large scale $s_{\text{max}}$, which we set to $1.0$ for all
experiments, corresponding to the extent of the input cameras’ positions.
These clusters form the top-level nodes in the scene decomposition.

Recursive Clustering: Next, to produce a hierarchical tree of scene nodes, we
iteratively reduce scale by a fixed epsilon (we use 0.05), running HDBSCAN on
each leaf node. If HDBSCAN returns more than one cluster for a given node, we
add those clusters as children and recurse. This continues until we reach
scale 0, at which point the procedure terminates, returning the current tree.

Figure 8: Segment-Anything [15] vs. GARField: SAM’s automatic mask generator
may encounter difficulty recalling all masks from a given viewpoint,
especially when there are clusters of small masks and the camera is far away
from the object. In contrast, GARField’s scale-conditioned affinity field
incorporates masks from multiple viewpoints in 3D.

##  5 Experiments

We assess GARField’s ability to decompose in-the-wild 3D scenes into
hierarchical groups which vary widely in size and semantics. Existing 3D scan
datasets tend to focus on object-level scans [7, 22], are simulated [2], or
contain primarily indoor household scenes [6]. To evaluate GARField, we
instead use a wide variety of indoor and outdoor scenes from the Nerfstudio
and LERF datasets, as well as additional captures for this paper. We
experiment on scenes which possess significant object hierarchy, testing the
decomposition ability of GARField. We provide qualitative results in Fig. 3
and Fig. 6, and quantitatively evaluate by annotating ground truth masks on
select scenes, a full list of which are in the Supplement.

Figure 9: Ablation: Without dense hierarchy supervision, points may have
inconsistent affinities across scales. There may be 1) spurious large
affinities at unsupervised scales or 2) unexpected drops in affinity at larger
scales.

###  5.1 Qualitative Scene Decomposition

We use Gaussian Splatting [12] to visualize the decomposition by querying
GARField’s affinity field at gaussian centers. We do this because gaussian
splats are easier to segment in 3D compared to NeRFs. See the Supplement for a
full description of the pipeline. All renderings are of complete 3D models,
not segmentations of 2D image views.

We visualize two types of hierarchical clustering results. In Fig. 7 we
globally cluster the scene at a hand selected coarse scale, then from these
scene-wide clusters we select groups corresponding to few objects and further
decompose them into subgroups. We visualize clusters obtained at successively
decreasing scales, which increases the granularity of groups. GARField
achieves high-fidelity 3D groupings across a wide range of scenes and objects,
from man-made objects – such as keyboards, where each key is considered a
group at a small scale, to the parts of the NERF gun and the Lego bulldozer –
to complex natural objects like plants, where it can group individual flowers
as well as their petal and leaves. By varying the scale, one can separate
objects at different levels, for instance the succulent from its pot versus
each individual leaf (first row), or identifying the bunny toy in the
bulldozer’s scooper, which is further grouped into its shirt, ears, and head
(fifth row, right). See Fig. 10 for select scene-wide cluster visualizations.

In Fig. 6 we visualize a tree decomposition produced by the method described
in Sec. 4. We first show the global clustering at a top level node, from which
we select the central statue to illustrate the tree decomposition. Arrows
denote children in the hierarchy, illustrating how the statues decomposes
gradually all the way down to its hair, legs, torso, etc. See the Supplement
for more tree visualizations.

| Fine | Medium | Coarse
---|---|---|---
Scene | SAM | Ours | SAM | Ours | SAM | Ours
teatime | 81.6 | 92.7 | 97.3 | 97.9 | - | -
bouquet | 17.4 | 76.0 | 73.5 | 81.6 | 76.1 | 85.4
keyboard | 65.3 | 88.8 | 73.6 | 98.4 | - | -
ramen | 53.3 | 79.2 | 74.7 | 90.7 | 92.6 | 95.5
living_room | 85.3 | 90.5 | 74.2 | 80.7 | 88.6 | 94.4

Table 1: 3D Completeness. We report mIOU of scene annotations for a single
point with up to three levels of hierarchy. SAM struggles to produce view-
consistent fine groups compared to GARField.

Scene | SAM [15] | Ours | Ours | Ours
---|---|---|---|---
(-scale) | (-dense)
ramen | 74.9 | 64.1 | 74.1 | 85.6
teatime | 64.9 | 67.7 | 66.1 | 86.6
keyboard | 23.2 | 57.6 | 73.1 | 77.9
bouquet | 34.4 | 49.8 | 72.9 | 76.4
living_room | 59.6 | 49.7 | 62.1 | 76.6

Table 2: Hierarchical Grouping Recall: We report mIOU against human
annotations of multi-scale groups of different objects.

###  5.2 Quantitative Hierarchy

We quantitatively evaluate our approach against annotated images using two
metrics: the first measuring view consistency against annotations from
multiple views and the second measuring recall of various hierarchical masks
via mIOU against ground truth human annotations.

3D Completeness: For downstream tasks it is useful for groups to correspond to
complete 3D objects, for example groups that contain an entire object rather
than just one of its sides. Though GARField always produces view-consistent
groups by construction, it may not necessarily contain complete objects. We
evaluate for completeness by checking that an entire 3D object is grouped
together across a range of viewpoints. To do this, on 5 scenes we choose a 3D
point to be projected into 3 different viewpoints, and label 3 corresponding
view-consistent ground truth masks containing that point at coarse, medium,
and fine levels. At these points we mine multiple masks from GARField across
multiple scales at 0.05 increments, where at each scale a mask is obtained
based on feature similarity thresholded at 0.9. We also compare against SAM by
clicking the point in the image and taking all 3 masks. We report the maximum
mIOU computed over all candidate masks for both methods.

Results are shown in Table 2. GARField produces more complete 3D masks than
SAM across viewpoints, resulting in higher mIOU with multi-view human
annotations of objects. This effect is especially apparent at the most
granular level, where from certain perspectives SAM struggles to produce fine
groups, like the keyboard keys from afar in Fig. 8. See the Supplement for
figures of comparisons and visualization of the groundtruth masks.

Hierarchical Grouping Recall: Here we measure GARField’s ability to recall
groups at multiple granularities. Across 5 scenes, we choose one novel
viewpoint and label up to 3 ground truth hierarchical groups for 1-2 objects.
GARField outputs a set of masks as described in Section 4 by clustering image-
space features, outputting one mask per tree node. We compare against SAM’s
automatic mask generation by keeping all output masks. We ablate GARField in
two ways: GARField (-scale) removes scale-conditioning; and GARField
(-hierarchy) removes the densified supervision in Sec. 3.2.2.

In Table 2 we report mIOU of the ground truth mask with the highest overlap,
either from the set of SAM masks or the tree generated by GARField. Because
GARField has fused groups from multiple perspectives, it results in higher
fidelity groupings than any single view of SAM, leading to higher mIOU with
annotations. Our ablations show that scale conditioning and scale
densification is necessary for high quality groupings. Fig. 9 illustrates
affinity degrading at higher scale with naive supervision.

##  6 Limitations

GARField at its core is distilling outputs from a 2D mask generator, so if the
masks fail to contain a desired group, this will not emerge in 3D. Regions
with uneven viewpoints can suffer from artificial group boundaries, for
example if an object is only viewed from close up, it may never be grouped
together because no input view contained it in full. We handle group ambiguity
using physical size, but there could be multiple groupings within a single
scale. For example, conflicts may happen with objects contained in a container
because the container with and without the object can have the same scale.
Future work could consider other ways to resolve grouping ambiguity such as
affordances. Another consequence of scale-conditioning is that object parts of
different sizes branch off the tree separately rather than all at once:
multiple objects on the same table may appear at different levels of the tree.
The tree generation in this work is a naive greedy algorithm, which can result
in spurious small groups at deeper levels, as seen in the trees in the
Supplement. Future work may explore more sophisticated ways of hierarchical
clustering.

Figure 10: Scene-Wide Clustering visualizations for selected scenes from Fig.
7.

##  7 Conclusion

We present GARField, a method for distilling multi-level masks into a dense
scale-conditioned affinity field for hierarchical 3D scene decomposition. By
leveraging scale-conditioning, the affinity field can learn meaningful groups
from conflicting 2D group inputs and break apart the scene at multiple
different levels, which can be used for extracting assets at a multitude of
granularities. GARField could have applications for tasks that require multi-
level groupings like robotics, dynamic scene reconstruction, or scene editing.

##  8 Acknowledgements

This project was funded in part by NSF:CNS-2235013 and DARPA Contract No.
HR001123C0021. Chung Min and Justin are supported in part by the NSF Graduate
Research Fellowship Program under Grant No. DGE 2146752. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the
National Science Foundation.

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Supplementary Material

##  Appendix A Additional Results

We show additional figures and videos using GARField for 1) hierarchical
decomposition, 2) global clustering, and 3) interactive selection. All video
visualizations use Gaussian Splatting [12], as described below.

###  A.1 Gaussian Splat Visualizations

We use Gaussian Splatting [12] to emphasize the 3D nature of GARField and its
applications for 3D group extraction. Here, for simplicity, we do not optimize
GARField directly with gaussians. Instead, we train a NeRF-based GARField and
a Gaussian Splatting model separately. Then, we assign an affinity feature to
every gaussian by querying the feature field at the gaussian’s center point.
We use these features to manipulate the 3D scene, _e.g_. clustering,
selection, and filtering. All implementation described here will be made
public. To visualize clusters in 3D, we override each gaussian’s color
parameters to the RGB color of the colormap.

###  A.2 3D Hierarchical Decomposition

In the main text, we visualized hand-picked nodes from the resulting hierarchy
in Main Paper Fig. 6. Here, we exhaustively visualize entire subtrees of
selected scenes by selecting the primary region of interest (i.e. desk, dozer,
bouquet).

####  A.2.1 Full Tree Visualizations

In Fig. 11 and in provided videos we visualize each layer of the resulting
tree organized by node depth in different rows. Each node is shown colorized
by the number of internal clusters, with the remainder of the tree drawn with
low opacity to give context. Note that nodes at the same level do not
necessarily correspond to the same scale because intermediate nodes are
pruned.

One can see how each part is recursively broken into subparts in lower layers
of the tree, for example the statue gets broken into the base and rest of the
statue, followed by shield, torso, hair, and etc. Note how some nodes can
contain noise or partial clusters, for example the third row, last node of
Fig. 11, where the red cluster is a spurious cluster which more suitably
belongs to the base of the statue in the prior tree level. Artifacts such as
this can happen as a result of our greedy tree building approach, and might be
addressed with a more sophisticated tree construction algorithm. Videos of
trees showcase the view-consistency of 3D scene decomposition, with whole
objects being clustered together like the bear or dozer, which can then be
broken into coherent subparts. The lowest levels of the tree contain very fine
details such as petals of flowers, or hooves of the sheep.

These exhaustive tree visualizations also exhibit limitations, such as
spurious background points being grouped together with the object of interest,
a behavior which could be remedied by more strongly taking geometric proximity
into account when constructing the tree. Another failure mode is that when
view coverage is insufficient, different sides of the same object can be
grouped separately. For example, in rows 3 and 4 of Fig. 11 the two sides of
the statue’s face are grouped differently.

Figure 11: Complete Tree: A complete visualization of all layers and all nodes
in the tree from Fig. 6. Colors illustrate different clusters within each
node, and each row visualizes all the nodes at a given depth in the tree,
sorted by size.

####  A.2.2 Compressed Tree Visualizations

We additionally provide videos of compressed trees, where each layer of the
tree is merged into one visual by distinctly coloring all clusters. Leaf nodes
at one layer are further propagated to deeper layers of the tree to visualize
all clusters at the lowest level, corresponding to the most granular
decomposition. Though these visualizations do not show hierarchy because they
merge all nodes, they illustrate how lower layers of the scene decomposition
correspond to semantically meaningful high granularity and higher levels
correspond to coarser granularities.

###  A.3 Multi-Scale Clustering

We provide video versions of Main Paper Fig. 7 to showcase the view-
consistency of the results shown in the images. These videos first show the
global clustering of the scene, followed by video renderings of sub-object
clusters.

###  A.4 Global Clustering

To emphasize that GARField can model scene-level groupings, we cluster
GARField features globally _i.e_. all gaussians in a scene. Figures 13 through
19 show all scenes in Fig. 7 globally clustered at scales 0 to 1, at
increments of 0.05.

We also include a video where the excavator scene in Main Paper Fig. 1 is
globally clustered at three distinct scales. We find that GARField
successfully groups together large group in the backgrounds, like the road or
bushes on the sidewalk.

###  A.5 Interactive Selection

People can use clicks to interact with GARField and extract groups of
different sizes, as shown in Fig. 5 of the main paper. User clicks are
transformed into 3D points using projective geometry (visualized with a red
sphere in the video). At a given scale, we select a set of 3D gaussians based
on their affinity with the selected point. To retrieve multiple groups, we
query GARField across a range of scales and merge groups with large overlap.
In the video, a user can extract the excavator, crane, and scooper from Fig. 1
with a single click.

##  Appendix B Experiment details

Figure 12: Masks for 3D Completeness Experiments: Overlapping masks (egg,
noodles, nori masks inside ramen mask) model the desired hierarchical
groupings. We labeled these polygonal masks using ‘Make Sense’ [30], an online
tool for mask annotation.

###  B.1 Hierarchical Decomposition

Once we select a cluster of interest, we construct a tree by recursively
clustering with HDBSCAN. For this process we use an HDBSCAN cluster epsilon of
0.1 and a minimum cluster size of 40, fixed for all experiments. The tree is
constructed greedily in a depth-first search, by recursing only on non-noise
clusters. Note that because we add noise clusters back to the tree after
constructing it, this can result in small disappearing regions, like in the
lower levels of the succulent scene. These artifacts would better be addressed
with a non-greedy tree construction, which we hope to address in future work.

To speed tree contruction, we first sub-sample the input gaussian splat with
Open3D’s voxel-downsampling to reduce the resolution of points to $0.01\times$
the scale being queried, for example an affinity of 0.1 scale downsamples to
.001 voxel resolution. After tree construction, the resulting tree is pruned
to remove chains of nodes with one child and one parent.

###  B.2 Treatment of Clustering Noise

One challenge to overcome is the fact that HDBSCAN can output ‘noise’
clusters, which do not get any cluster labels. These can arise because of
gaussians which do not align well with NeRF geometry, features which are noisy
because they lie on the boundary of two groups, or noise in the trained
affinity field. To handle these noise clusters, we assign labels to gaussians
considered noise with the label of the nearest physical clusters computed in
the Euclidean space, as opposed to the feature space. We find this produces
more cohesive results than soft clustering within the feature space itself.
During global clustering (Figs. 10, 7) these noise clusters are assigned to
custers across the entire scene, and during tree decomposition (Fig. 6) these
noise clusters are locally assigned from the clusters available at each node
only.

###  B.3 3D Completeness Experiment

####  B.3.1 Ground Truth Annotation

We annotate ground truth segmentation masks on a randomly selected novel view
using the online tool ‘Make Sense’ [30], employing a polygon shape for the
annotation. In Fig. 12, we present the visualization on our state during the
data annotation process.

The annotation process begins with the assignment of a specific label point to
each target object within a given view. Note that the selection of the view is
randomized, involving zooming in, zooming out, or changing the angle to
enhance the evaluation of view consistency effectively. These label points
serve as the basis for the subsequent mask annotation, which are made at a
varying level of granularity. As a case in Fig. 21, in bouquet scene,
considering the click points from different angles, we annotate the masks at
different hierarchical levels: the petal of the flower (fine level), the
individual flower (medium level) and the whole bouquet (coarse level). For
ground truth masks in other scenes, we follow similar rules, building a mask
hierarchy based on the semantic meaning, ranging from fine part of the object
to coarse whole object. However, note that the number of mask levels may vary
depending on the complexity and the nature semantics in the scene. For
example, the bear’s arm in the teatime scene, Fig. 20, is only annotated with
two levels of hierarchy: the left hand and the whole bear.

####  B.3.2 Complete Visualizations

A comprehensive presentation of the evaluation results regarding to the view
consistency of GARField is shown in Figs. 20, 21, 22, 23, 24. This includes
all the scenes not shown in the main text. For each scene, we show the clicked
label points for the annotated randomly selected views, ground truth masks at
different hierarchical levels and the comparison of the closest masks obtained
by SAM and GARField. We also provide the zoomed-in images of the results for
better visualization.

###  B.4 Hierarchical Grouping Recall Experiment

####  B.4.1 Ground Truth Annotation

In this experiment, we annotate one novel view for each of the five scenes.
For each novel view, we mark one or several objects which has a rich
hierarchy. The ground truth masks are any parts, subparts, or the entire
object of the scene that can be considered as groups by a human. Taking the
ramen scene (Figs. 12, 25) as an example, the parts or subparts of the objects
labeled include nori, egg ,egg yolk, noodles, and so on. Additionally, the
complete soup and the entire ramen bowel is also annotated as a group. Unlike
the experiments on 3D completeness, this experiment aims to test whether the
model can extract all the reasonable masks of the objects which contain rich
hierarchy. Therefore, we did not stratify the level of the annotated masks.

####  B.4.2 Complete Visualization

In Fig. 25, We show the ground truth masks as well as all the methods masks at
the finest masks. Note that all the ground truth masks are arranged in
descending order of size. In our experiment, we systematically recover all the
masks that corresponds to the annotated ground truth through different method.
For each distinct method employed, which are SAM, GARField without scale
condition, GARField without dense supervision, we sequentially showcase the
masks that get the highest IOU score of the correspondence to the ground truth
masks. We will release all the ground truth annotations for all experiments.

Figure 13: Global Clustering Results (“Bouquet”): Global clusters at smaller
scales $(s=0)$ distinguish between different sections of the bouquet, as well
as the two halves of the table. At a larger scale, the bouquet and table are
considered whole. Figure 14: Global Clustering Results (“Desk”): At larger
scales $(s=0.5)$, the desk is grouped together with the clutter on it _e.g_.
keyboard, card, bird figurine). Figure 15: Global Clustering Results
(“Donuts”): At a very small scale $(s=0.0)$, GARField can distinguish between
different pieces of the breakfast sandwich in the middle of the scene. As
scale increases, its grouping shifts quite noticably — into its two halves, or
the full sandwich with the checkerboard packaging. Figure 16: Global
Clustering Results (“Table”): At the smallest scale $(s=0.0)$, the global
clusters highlight parts of objects _e.g_. labels on water bottles, pieces of
chocolate. Figure 17: Global Clustering Results (“Teatime”): The food,
utensils, and the table are included in different clusters at small scales,
and the same cluster at larger scales. Parts of the stuffed animals (_e.g_.
sheep hooves, bear nose) can also be seen at $s=0.0$. Figure 18: Global
Clustering Results (“Succulent”): Global clusters at smaller scales ($s=0.0$)
distinguish between fine features like succulent leaves, while they are
considered a single group at larger scales ($s=1.0$). Figure 19: Global
Clustering Results (“Living Room”: The individual hexagonal tiles on the floor
may be grouped separately ($s=0.0$) or together $(s=0.5)$.  Figure 20: View
Consistency Experiment-Teatime: We constructed two hierarchies, which are fine
and medium. These correspond to the semantic meanings of the bear’s left hand
and the whole bear, respectively. Figure 21: View Consistency Experiment-
Bouquet: We constructed three hierarchies, which are fine medium and coarse.
These correspond to the semantic meanings of the petal of the flower, the
individual flower and the whole bouquet, respectively. Figure 22: View
Consistency Experiment-Keyboard: We constructed two hierarchies, which are
fine and medium. These correspond to the semantic meanings of single key and
the whole keyboard, respectively. Figure 23: View Consistency Experiment-
Ramen: We constructed three hierarchies, which are fine, medium and coarse.
These correspond to the semantic meanings of egg yolk , one single egg and the
whole soup area, respectively. Figure 24: View Consistency Experiment-Living
room: We constructed two hierarchies, which are fine medium and coarse. These
correspond to the semantic meanings of the small orange part of the nerf gun,
medium blue part of the nerf gun and the whole nerf gun, respectively. Figure
25: Hierarchical Grouping Recall Experiments: We concentrate on methods such
as SAM and the ablation study of GARField. GARField outperforms SAM in
obtaining finer, smaller masks (e.g. capturing all the tiny keys in a keyboard
scene). Unlike GARField without hierarchy grouping, GARField achieves more
layered grouping results (e.g. in the ramen scene, it successfully identifies
the entire ramen mask through hierarchical clustering). Furthermore, compared
to GARField without dense supervision, GARField provides more stable and
thorough grouping outcomes (e.g. in the teatime scene, GARField more
comprehensively identifies the small labels on the cookie bag).