# Evidence of the low-lying baryon $\Sigma^{*}(1/2^{-})$ in the process
$\Lambda_{c}^{+}\to\eta\pi^{+}\Lambda$

Wen-Tao Lyu  School of Physics, Zhengzhou University, Zhengzhou 450001, China
Sheng-Chao Zhang  Joint Research Center for Theoretical Physics, School of
Physics and Electronics, Henan University, Kaifeng 475004, China  Guan-Ying
Wang<EMAIL_ADDRESS>Joint Research Center for Theoretical Physics,
School of Physics and Electronics, Henan University, Kaifeng 475004, China
Jia-Jun Wu<EMAIL_ADDRESS>School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Southern Center for
Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of
Sciences, Huizhou 516000, China  En Wang<EMAIL_ADDRESS>School of Physics,
Zhengzhou University, Zhengzhou 450001, China  Guangxi Key Laboratory of
Nuclear Physics and Nuclear Technology, Guangxi Normal University, Guilin
541004, China  Li-Sheng Geng<EMAIL_ADDRESS>School of Physics,
Beihang University, Beijing 102206, China  Peng Huanwu Collaborative Center
for Research and Education, Beihang University, Beijing 100191, China  Beijing
Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University,
Beijing 102206, China  Southern Center for Nuclear-Science Theory (SCNT),
Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000,
China  Ju-Jun Xie<EMAIL_ADDRESS>Southern Center for Nuclear-Science
Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences,
Huizhou 516000, China  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China

###### Abstract

Motivated by the Belle measurements of the process
$\Lambda_{c}^{+}\to\eta\pi^{+}\Lambda$, we investigate this process by
considering the contributions from the $\Lambda(1670)$, $a_{0}(980)$, and
$\Sigma(1385)$. In addition, we also consider the predicted low-lying baryon
$\Sigma^{*}(1/2^{-})$. Our results involving the $\Sigma^{*}(1/2^{-})$ are
favored by fitting to the Belle data of the $\eta\Lambda$ and $\pi^{+}\Lambda$
invariant mass distributions. Furthermore, we predict the $\eta\pi^{+}$
invariant mass distribution and the angular distribution $d\Gamma/d{\rm
cos}\theta$, which are significantly different depending on whether or not the
contribution from the $\Sigma^{*}(1/2^{-})$ is considered. Finally, we show
that, with the contribution from the $\Sigma^{*}(1/2^{-})$, the calculated
Dalizt plot agrees with the Belle measurements. Future precise measurements of
the process $\Lambda_{c}^{+}\to\eta\pi^{+}\Lambda$ could shed further light on
the existence of the low-lying $\Sigma^{*}(1/2^{-})$.

##  I Introduction

Since the charmonium-like state $X(3872)$ was observed in the
$\pi^{+}\pi^{-}J/\psi$ invariant mass distribution of the process $B^{\pm}\to
K^{\pm}\pi^{+}\pi^{-}J/\psi$ by the Belle Collaboration in 2003 [1], a
plethora of hadrons beyond the conventional quark model have been observed
experimentally. Exploring their nature is crucial to understanding the non-
perturbative properties of Quantum Chromodynamics (QCD) [2, 3, 4, 5, 6, 7]. It
should be stressed that most of these exotic hadrons are located near the mass
thresholds of pairs of hadrons, and many theoretical studies have concluded
that the hadron-hadron interactions play an essential role in these exotic
hadrons [4, 5, 6].

As we know, the low-lying baryon $\Lambda(1405)$ with $J^{P}=1/2^{-}$ could be
generated from the $\bar{K}N$ interaction in the unitary chiral approach [8,
9]. However, the low-lying excited $\Sigma^{*}(1/2^{-})$ has not yet been well
established experimentally and theoretically [11]. Although the state
$\Sigma(1620)$ with $J^{P}=1/2^{-}$ is listed in the Review of Particle
Physics (RPP) [11], its status is only one star (meaning that the evidence of
its existence is poor) and is omitted from the summary tables, which implies
that this state still needs to be confirmed. Furthermore, the
$\Sigma^{*}(1/2^{-})$ with a mass around 1.4 GeV as a partner of
$\Lambda(1405)$ is not found yet. Thus, searching for the state
$\Sigma^{*}(1/2^{-})$ is crucial to deepening our understanding of low-lying
excited baryons [12, 13, 5], and many theoretical works have been devoted to
this issue.

For instance, the $S$-wave meson-baryon interaction in the $S=-1$ sector was
studied in the unitary chiral approach, and one $\Sigma^{*}(1/2^{-})$ with a
mass around the $\bar{K}N$ threshold was predicted in Refs. [14, 15, 8, 16,
17, 9, 18, 19, 20], which is supported by the analysis of the CLAS data on the
process $\gamma p\to K\Sigma\pi$ [22, 21]. By analysing the relevant data of
$K^{-}p\to\Lambda\pi^{+}\pi^{-}$, Refs. [24, 23] suggested that there may
exist one $\Sigma^{*}(1/2^{-})$ resonance with a mass about 1380 MeV. In the
effective Lagrangian approach, Ref. [25] found that the $\Sigma^{*}(1/2^{-})$
plays an important role in the $K\Sigma^{*}(1385)$ photoproduction. In
addition, it is also suggested to search for the $\Sigma^{*}(1/2-)$ in the
processes of $\Lambda_{c}\to\eta\Lambda\pi^{+}$ [26],
$\chi_{c0}(1P)\to\bar{\Lambda}\Sigma\pi$ [27],
$\chi_{c0}(1P)\to\bar{\Sigma}\Sigma\pi$ [28], and $\gamma N\to K\Sigma^{*}$
[29, 30]. Recently, two $\Sigma^{*}(1/2^{-})$ states were predicted in the
global study of meson-baryon scattering in the covariant baryon chiral
perturbation theory, including the next-to-next-to-leading order
contributions, where the narrower pole located at $(1432,-i18)$ MeV shows up
as a cusp, and the broader one located at $(1364,-i110)$ MeV becomes a broad
enhancement in the real axis of the amplitude squared [31].

Based on a 980 fb-1 data sample, the Belle Collaboration measured the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$, which had been measured by CLEO [32]
and BESIII [33] earlier, and presented the $\eta\Lambda$ and $\pi^{+}\Lambda$
invariant mass distributions, which show significant signals of the resonances
$\Lambda(1670)$, $a_{0}(980)$, and $\Sigma^{*}(1385)$ [34]. In Ref. [35], we
have analyzed this process by considering the intermediate state
$\Sigma^{*}(1385)$ ($J^{P}=3/2^{+}$), and the $S$-wave $\eta\Lambda$ and
$\eta\pi^{+}$ final-state interactions in the chiral unitary approach, which
dynamically generate the $\Lambda(1670)$ and $a_{0}(980)$ resonances. Our
results could reproduce the Belle data of the $\pi^{+}\Lambda$ and
$\eta\Lambda$ invariant mass distributions. However, the Dalitz plot of the
process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ has more yields appearing in
the region of $M^{2}_{\Lambda\pi^{+}}<1.85$ GeV2 and $M^{2}_{\eta\Lambda}<4$
GeV2, which cannot be well described in Ref. [35]. This discrepancies may
imply some contributions from the resonance $\Sigma^{*}(1/2^{-})$ [26]. In
principle, the signal of $\Sigma^{*}(1/2^{-})$ could overlap with the one of
$\Sigma(1385)$, and will be difficult to be directly observed in the
$\Lambda\pi^{+}$ invariant spectrum. However, the contributions from
$\Sigma^{*}$ is crucial to understanding the invariant mass spectrum of
$\pi^{+}\eta$ correctly. Thus, the invariant mass distribution of
$\pi^{+}\eta$ is essential for us to determine the existence of
$\Sigma^{*}(1/2^{-})$, but it is missing in the publication of the Belle
Collaboration [34].

In this work, besides the intermediate $\Sigma(1385)$, the $S$-wave final-
state interactions of the $\eta\Lambda$ and $\eta\pi^{+}$, we also consider
the contribution from the intermediate $\Sigma^{*}(1/2^{-})$ in the process of
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ and show the evidence of this state by
analyzing the Belle measurements.

This paper is organized as follows. In Sec. II, we present the theoretical
formalism for the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$. Numerical
results are presented in Sec. III. Finally, we present a summary in the last
section.

##  II Formalism

This section will introduce the theoretical formalism for the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$. First, in Sec. II.1, we present the
mechanisms for this process via the $S$-wave meson-baryon (MB) interactions,
which would generate the state $\Lambda(1670)$. Second, in Sec. II.2, we
present the mechanisms for this process via the meson-meson (MM) interactions,
which would generate the scalar $a_{0}(980)$. The contributions from the
intermediate states $\Sigma(1385)$ and $\Sigma^{*}(1/2^{-})$ are discussed in
Sec. II.3. Finally, we present the formalism for the invariant mass
distributions and angular distributions of the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ in Sec. II.4.

###  II.1 $S$-wave meson-baryon ($MB$) interactions and $\Lambda(1670)$

Figure 1: Quark-level diagram for the
$\Lambda^{+}_{c}\to\pi^{+}s(\bar{u}u+\bar{d}d+\bar{s}s)ud$ process.

For the process $\Lambda_{c}^{+}\to\pi^{+}(MB)\to\pi^{+}\eta\Lambda$, the
dominant mechanism is the $W^{+}$ external emission, as shown in Fig. 1. The
$c$ quark of the initial $\Lambda^{+}_{c}$ will firstly weakly decay into an
$s$ quark and a $W^{+}$ boson, then the $W^{+}$ boson subsequently decays into
a $u\bar{d}$ quark pair. The $u\bar{d}$ quark from the $W^{+}$ boson will
hadronize into a $\pi^{+}$ meson of high momentum, while the $s$ quark from
the initial $\Lambda^{+}_{c}$ and the $ud$ quark pair of the initial
$\Lambda^{+}_{c}$, together with the quark pair
$\bar{q}q=\bar{u}u+\bar{d}d+\bar{s}s$ created from the vacuum with the quantum
numbers $J^{PC}=0^{++}$, hadronize into hadron pairs, which could be expressed
as,

$\displaystyle\Lambda_{c}^{+}$ $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}c(ud-
du)\Rightarrow\frac{1}{\sqrt{2}}W^{+}s(ud-du)$ (1) $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}u\bar{d}s(\bar{u}u+\bar{d}d+\bar{s}s)(ud-du)$
$\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}\pi^{+}s(\bar{u}u+\bar{d}d+\bar{s}s)(ud-du)$
$\displaystyle\Rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}\pi^{+}\sum
M_{3i}q_{i}(ud-du).$

Here $M$ is the matrix of the pseudoscalar mesons,

$\displaystyle
M=\left(\begin{matrix}\frac{{\eta}}{\sqrt{3}}+\frac{{\pi}^{0}}{\sqrt{2}}+\frac{{\eta}^{\prime}}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{{\eta}}{\sqrt{3}}-\frac{{\pi}^{0}}{\sqrt{2}}+\frac{{\eta}^{\prime}}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\sqrt{\frac{2}{3}}{\eta}^{\prime}-\frac{{\eta}}{\sqrt{3}}\end{matrix}\right),$
(2)

where we have considered the approximate $\eta-\eta^{\prime}$ mixing [36, 37].

Then, we have the components of the pseudoscalar meson and octet baryon as,

$\displaystyle\frac{1}{\sqrt{2}}\sum M_{3i}q_{i}(ud-du)$ $\displaystyle=$
$\displaystyle K^{-}\left[\frac{1}{\sqrt{2}}u(ud-
du)\right]+\bar{K}^{0}\left[\frac{1}{\sqrt{2}}d(ud-
du)\right]-\frac{{\eta}}{\sqrt{3}}\left[\frac{1}{\sqrt{2}}s(ud-du)\right]$ (3)
$\displaystyle=$ $\displaystyle
K^{-}p+\bar{K}^{0}n+\frac{\sqrt{2}}{3}\eta\Lambda,$

where we have ignored the $\eta^{\prime}\Lambda$ channel since its threshold
($2073.5$ MeV) is far from the mass region of $\Lambda(1670)$. Here, we take
the flavor wave functions for the octet baryons as [39, 40, 38],

$\displaystyle p=\frac{u(ud-du)}{\sqrt{2}},~{}~{}~{}n=\frac{d(ud-
du)}{\sqrt{2}},$ (4) $\displaystyle\Lambda=\frac{u(ds-sd)+d(su-us)-2s(ud-
du)}{2\sqrt{3}}.$ (5)

Figure 2: The mechanisms for the $\Lambda_{c}\to\pi^{+}\eta\Lambda$ decay. (a)
tree diagram, (b) the $S$-wave meson-baryon interactions.

Then, the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay could occur via
the tree diagram of Fig. 2(a), and the $S$-wave meson-baryon interaction of
Fig. 2(b). The decay amplitude could be expressed as,

$\displaystyle\mathcal{T}^{MB}(M_{\eta\Lambda})=$ $\displaystyle
V_{p}\left[\frac{\sqrt{2}}{3}+G_{K^{-}p}(M_{\eta\Lambda})t_{K^{-}p\to\eta\Lambda}(M_{\eta\Lambda})\right.$
(6)
$\displaystyle+G_{\bar{K}^{0}n}(M_{\eta\Lambda})t_{\bar{K}^{0}n\to\eta\Lambda}(M_{\eta\Lambda})$
$\displaystyle\left.+\frac{\sqrt{2}}{3}G_{\eta\Lambda}(M_{\eta\Lambda})t_{\eta\Lambda\to\eta\Lambda}(M_{\eta\Lambda})\right],$

where $G_{l}$ are the loop functions of the meson-baryon system, and the
parameter $V_{p}$, accounting for the weak decay and hadronization strength,
is approximately independent of the final-state interactions [41, 5]. The
$t_{MB\to\eta\Lambda}$ are the transition amplitudes, which can be obtained by
solving the Bethe-Salpeter equation,

$T=[1-VG]^{-1}V.$ (7)

The transition potential $V_{ij}$ is taken from Ref. [14],

$\displaystyle V_{ij}$
$\displaystyle=-C_{ij}\frac{1}{4f^{2}}(2\sqrt{s}-M_{i}-M_{j})$ (8)
$\displaystyle\times\left(\dfrac{M_{i}+E_{i}}{2M_{i}}\right)^{1/2}\left(\dfrac{M_{j}+E_{j}}{2M_{j}}\right)^{1/2},$

where $E_{i}$ and $M_{i}$ are the energy and mass of the baryon in the $i$-th
channel, and the coefficients $C_{ij}$ reflect the $SU(3)$ flavor symmetry,
and are given in Table I of Ref. [8]. $\sqrt{s}$ is the invariant mass of the
$\eta\Lambda$ system. The coupling $f$ is the pseudoscalar decay constant, for
which we use,

$f=1.15f_{\pi},~{}~{}f_{\pi}=93~{}\mathrm{MeV}.$ (9)

The $G_{l}$ can be written by [14],

$\displaystyle G_{l}=$ $\displaystyle
i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{M_{l}}{E_{l}(\vec{q}\,)}\frac{1}{\sqrt{s}-q^{0}-E_{l}(\vec{q}\,)+i\epsilon}\frac{1}{q^{2}-m_{l}^{2}+i\epsilon}$
(10) $\displaystyle=$
$\displaystyle\frac{2M_{l}}{16\pi^{2}}\left\\{a_{l}(\mu)+\ln\frac{M_{l}^{2}}{\mu^{2}}+\frac{s+m_{l}^{2}-M_{l}^{2}}{2s}\ln\frac{m_{l}^{2}}{M_{l}^{2}}\right.$
$\displaystyle+\frac{|\vec{q}\,|}{\sqrt{s}}\left[\ln\left(s-\left(M_{l}^{2}-m_{l}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)\right.$
$\displaystyle+\ln\left(s+\left(M_{l}^{2}-m_{l}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)$
$\displaystyle-\ln\left(-s+\left(M_{l}^{2}-m_{l}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)$
$\displaystyle\left.\left.-\ln\left(-s-\left(M_{l}^{2}-m_{l}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)\right]\right\\},$

where $m_{l}$ and $M_{l}$ are the masses of meson and baryon of the $l$-th
channel. Here, we take $\mu=630$ MeV. Since the pole position of
$\Lambda(1670)$ is quite sensitive to the value of $a_{K\Xi}$ and only
moderately sensitive to $a_{\bar{K}N}$, $a_{\pi\Sigma}$, $a_{\eta\Lambda}$, we
take $a_{K\Xi}$ to be a free parameter, and adopt the values
$a_{\bar{K}N}=-1.84$, $a_{\pi\Sigma}=-2.00$, and $a_{\eta\Lambda}=-2.25$ from
Ref. [14].

###  II.2 $S$-wave meson-meson ($MM$) interactions and $a_{0}(980)$

Figure 3:  Quark level diagram for the $\Lambda^{+}_{c}\to\Lambda
u(\bar{u}u+\bar{d}d+\bar{s}s)\bar{d}$ process.

The primary mechanism for the process $\Lambda_{c}^{+}\to
MM\Lambda\to\pi^{+}\eta\Lambda$ is the $W^{+}$ external emission, as shown in
Fig. 3. The $c$ quark of the initial $\Lambda^{+}_{c}$ weakly decays into an
$s$ quark and a $W^{+}$ boson, and then the $W^{+}$ boson subsequently decays
into a $u\bar{d}$ quark pair. The $s$ quark from the $\Lambda^{+}_{c}$ decay
and the $ud$ quark pair of the initial $\Lambda^{+}_{c}$ will hadronize into a
$\Lambda$ baryon, while the $u\bar{d}$ quark pair from the $W^{+}$ boson,
together with the quark pair $\bar{q}q=\bar{u}u+\bar{d}d+\bar{s}s$ created
from the vacuum with the quantum numbers $J^{PC}=0^{++}$, hadronize into
hadron pairs,

$\displaystyle\Lambda_{c}^{+}$ $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}c(ud-du)$ (11) $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}W^{+}s(ud-du)$ $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}u\bar{d}s(ud-du)$ $\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}u(\bar{u}u+\bar{d}d+\bar{s}s)\bar{d}s(ud-du)$
$\displaystyle\Rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}\sum
M_{1i}M_{i2}s(ud-du)$ $\displaystyle\Rightarrow$
$\displaystyle\left(\frac{2}{\sqrt{3}}\pi^{+}\eta+K^{+}\bar{K}^{0}\right)\frac{\sqrt{6}}{3}\Lambda$
$\displaystyle=$
$\displaystyle\left(\frac{2\sqrt{2}}{3}\pi^{+}\eta+\frac{\sqrt{6}}{3}K^{+}\bar{K}^{0}\right)\Lambda.$

Figure 4: The mechanisms for the $\Lambda^{+}_{c}\to\pi^{+}\eta\Lambda$ decay.
(a) tree diagram, (b) the $S$-wave pseudoscalar meson-meson interactions

Then, the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay occurs via the
tree diagram of Fig. 4(a), and the $S$-wave pseudoscalar meson-pseudoscalar
meson interaction of Fig. 4(b). The amplitude could be expressed as,

$\displaystyle\mathcal{T}^{MM}=$ $\displaystyle
V^{\prime}_{p}\left[\frac{2\sqrt{2}}{3}+\frac{2\sqrt{2}}{3}\tilde{G}_{\pi^{+}\eta}t_{\pi^{+}\eta\to\pi^{+}\eta}\right.$
(12)
$\displaystyle\left.+\frac{\sqrt{6}}{3}\tilde{G}_{K^{+}\bar{K}^{0}}t_{K^{+}\bar{K}^{0}\to\pi^{+}\eta}\right],$

where $\tilde{G}_{MM}$ in Eq. (12) is the loop function of the meson-meson
system, and $t_{MM\to\pi^{+}\eta}$ are the scattering matrices of the coupled
channels. The transition amplitude of $t_{MM\to\pi^{+}\eta}$ is obtained in
the chiral unitary approach. The details can be found in Ref. [35]. Here,
$V^{\prime}_{p}$ accounts for the strength of the weak vertex and
hadronization. Note that the tree diagrams of Figs. 2(a) and 4(a) are similar,
but their vertexes are different. In this work, we take
$V^{\prime}_{p}=0.155V_{p}$, which is obtained assuming that the amplitudes of
Eq. (6) and Eq. (12) give rise to the same width for the process
$\Lambda_{c}\to\eta\pi^{+}\Lambda$ [42]. In this work, the loop function
$\tilde{G}_{l}$ in Eq. (12) is calculated using the cutoff method,

$\displaystyle\tilde{G}_{l}$
$\displaystyle=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{(p-q)^{2}-m_{2}^{2}+i\epsilon}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}$
(13) $\displaystyle=\int_{0}^{q_{\rm
max}}\frac{d^{3}q}{(2\pi)^{3}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}\frac{1}{s-(\omega_{1}+\omega_{2})^{2}+i\epsilon},$

with $\omega_{1/2}=\sqrt{m^{2}_{1/2}+\vec{q}^{\,2}}$. In this work, we take
the cutoff momentum $q_{\rm max}=600$ MeV [43, 44, 45]. It should be stressed
that we use two different methods to derive the $G$ function, as shown in Eqs.
(10) and (13); this is because in the two systems (meson-baryon and meson-
meson), the existing works can help reduce the free parameters in this work .

###  II.3 Intermediate states $\Sigma(1385)$ and $\Sigma^{*}(1/2^{-})$

Figure 5: The contribution from the intermediate $\Sigma^{*}$. Figure 6: The
angle $\theta$ between $\pi^{+}$ and $\eta$ in the center of mass frame of the
$\pi^{+}\Lambda$ rest frame.

For the process $\Lambda^{+}_{c}\to\eta\Sigma(1385)\to\eta\pi^{+}\Lambda$
depicted in Fig. 5, the amplitude can be written as follows [46],

$\mathcal{T}^{\Sigma(1385)}=\dfrac{V^{\Sigma(1385)}|p^{*}_{\pi^{+}}||p^{*}_{\eta}|{\rm
cos}\theta}{M^{2}_{\pi^{+}\Lambda}-M^{2}_{\Sigma(1385)}+iM_{\Sigma(1385)}\Gamma_{\Sigma(1385)}},$
(14)

where $V^{\Sigma(1385)}$ is the relative strength of the contribution from the
intermediate resonance $\Sigma(1385)$, and $|p^{*}_{\pi}|$ and
$|p^{*}_{\eta}|$ are the momenta of $\pi^{+}$ and $\eta$ in the rest frame of
the $\pi^{+}\Lambda$ system, respectively, and $\theta$ is the angle between
$\pi^{+}$ and $\eta$ in the center of mass frame of the $\pi^{+}\Lambda$
system, as depicted in Fig. 6, which are given by [46],

$|p^{*}_{\pi^{+}}|=\dfrac{\lambda^{1/2}(M_{\pi^{+}\Lambda}^{2},m_{\pi^{+}}^{2},M_{\Lambda}^{2})}{2M_{\pi^{+}\Lambda}},$
(15)
$|p^{*}_{\eta}|=\dfrac{\lambda^{1/2}(M_{\Lambda_{c}}^{2},m_{\eta}^{2},M_{\pi^{+}\Lambda}^{2})}{2M_{\pi^{+}\Lambda}},$
(16) ${\rm
cos}\theta=\dfrac{M_{\eta\Lambda}^{2}-M_{\Lambda^{+}_{c}}^{2}-m_{\pi^{+}}^{2}+2P_{\Lambda^{+}_{c}}^{0}P_{\pi^{+}}^{0}}{2|\vec{p}_{\pi^{+}}||\vec{p}_{\eta}|},$
(17)

with the K$\ddot{a}$llen function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$. In the $\pi^{+}\Lambda$ rest
frame, $\vec{p}_{\Lambda^{+}_{c}}=\vec{p}_{\eta}$,
$\vec{p}_{\pi^{+}}=-\vec{p}_{\Lambda}$, and the $\Lambda_{c}$ and $\pi^{+}$
energies are,

$\displaystyle P_{\Lambda^{+}_{c}}^{0}$
$\displaystyle=\sqrt{M_{\Lambda^{+}_{c}}^{2}+|\vec{p}_{\Lambda^{+}_{c}}|^{2}}=\sqrt{M_{\Lambda^{+}_{c}}^{2}+|\vec{p}_{\eta}|^{2}},$
(18) $\displaystyle P_{\pi^{+}}^{0}$
$\displaystyle=\sqrt{m_{\pi^{+}}^{2}+|\vec{p}_{\pi^{+}}|^{2}}.$

As we discussed in the introduction, besides the $\Sigma(1385)$, we will also
take into account the contribution from the resonance $\Sigma^{*}(1/2^{-})$ in
the process $\Lambda^{+}_{c}\to\eta\Sigma^{*}(1/2^{-})\to\eta\pi^{+}\Lambda$,
as shown in Fig. 5. The amplitude of Fig. 5 can be written as follows,

$\mathcal{T}^{\Sigma^{*}(1/2^{-})}=\dfrac{V^{\Sigma^{*}(1/2^{-})}M_{\Sigma^{*}(1/2^{-})}\Gamma_{\Sigma^{*}(1/2^{-})}}{M^{2}_{\pi^{+}\Lambda}-M^{2}_{\Sigma^{*}(1/2^{-})}+iM_{\Sigma^{*}(1/2^{-})}\Gamma_{\Sigma^{*}(1/2^{-})}},$
(19)

where $V^{\Sigma^{*}(1/2^{-})}$ is the relative strength of the contribution
from the intermediate resonance $\Sigma^{*}(1/2^{-})$. According to the Belle
measurements of the $\pi^{+}\Lambda$ invariant mass distribution, if the
$\Sigma^{*}(1/2^{-})$ exists, its mass should be close to that of
$\Sigma(1385)$. Thus, in this work, we take $M_{\Sigma^{*}(1/2^{-})}=1380$ MeV
and $\Gamma_{\Sigma^{*}(1/2^{-})}=120$ MeV from Refs. [24, 23, 27].

###  II.4 Invariant Mass Distributions

With all the ingredients obtained in the previous section, one can write down
the modulus squared of the total amplitude as,

$\displaystyle|\mathcal{T}^{\text{Total}}|^{2}=$
$\displaystyle|\mathcal{T}^{MM}+\mathcal{T}^{MB}e^{i\phi}+\mathcal{T}^{\Sigma(1385)}e^{i\phi^{\prime}}$
(20)
$\displaystyle+\mathcal{T}^{\Sigma^{*}(1/2^{-})}e^{i\phi^{\prime\prime}}|^{2},$

where the $\phi$, $\phi^{\prime}$, and $\phi^{\prime\prime}$ are the phase
angles between different contributions. Then, the invariant mass distribution
of the double differential width of the $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$
can be written as,

$\displaystyle\dfrac{d^{2}\Gamma}{dM_{\pi^{+}\eta}^{2}dM_{\eta\Lambda}^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\dfrac{M_{\Lambda}}{8M_{\Lambda^{+}_{c}}^{2}}|\mathcal{T}^{\text{Total}}|^{2},$
(21)
$\displaystyle\dfrac{d^{2}\Gamma}{dM_{\pi^{+}\eta}^{2}dM_{\pi^{+}\Lambda}^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\dfrac{M_{\Lambda}}{8M_{\Lambda^{+}_{c}}^{2}}|\mathcal{T}^{\text{Total}}|^{2}.$
(22)

For a given value of $M_{12}$, the range of $M_{23}$ is determined by [11],

$\displaystyle\left(m_{23}^{2}\right)_{\min}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}+\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
$\displaystyle\left(m_{23}^{2}\right)_{\max}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}-\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
(23)

where $E_{2}^{*}$ and $E_{3}^{*}$ are the energies of particles 2 and 3 in the
$M_{12}$ rest frame, which are written as

$\displaystyle E_{2}^{*}=\dfrac{M_{12}^{2}-m_{1}^{2}+m_{2}^{2}}{2M_{12}},$
$\displaystyle
E_{3}^{*}=\dfrac{M_{\Lambda^{+}_{c}}^{2}-M_{12}^{2}-m_{3}^{2}}{2M_{12}},$ (24)

where $m_{1}$, $m_{2}$, and $m_{3}$ are the masses of particles 1, 2, and 3,
respectively.

On the other hand, the angular distribution of the differential decay width in
$d\Gamma/d{\rm cos}\theta$ is,

$d^{2}\Gamma=\frac{1}{(2\pi)^{3}}\dfrac{M_{\Lambda}}{4M_{\Lambda_{c}}}|p_{\eta}||p^{*}_{\pi^{+}}||\mathcal{T}^{\text{Total}}|^{2}dM_{\pi\Lambda}d{\rm
cos}\theta,$ (25)

where $\theta$ is the angle of the outing $\Lambda$ (or $\pi^{+}$) in the
center-of-mass frame of the $\pi^{+}\Lambda$ system, defined in Eq. (17), and
$|p^{*}_{\pi^{+}}|$ is the momenta of $\pi^{+}$ in the rest frame of the
$\pi^{+}\Lambda$ system (Eq. (15)), $|p_{\eta}|$ is the momenta of $\eta$ in
the rest frame of the $\Lambda^{+}_{c}$,

$|p_{\eta}|=\dfrac{\lambda^{1/2}(M_{\Lambda^{+}_{c}}^{2},m_{\eta}^{2},M_{\pi^{+}\Lambda}^{2})}{2M_{\Lambda^{+}_{c}}}.$
(26)

The particles’ masses and widths are taken from the RPP [11].

##  III Numerical results and discussions

Figure 7: Modulus squared of the transition amplitudes
$|T_{K\Xi\to\eta\Lambda}|^{2}$.

Figure 8: The $\eta\Lambda$ invariant mass distributions of the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay for (a) Fit A: without the
contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the
contribution from the $\Sigma^{*}(1/2^{-})$. The magenta-dashed curves show
the contribution from the meson-baryon interaction (MB), the blue-dotted
curves show the contribution from the meson-meson interaction (MM), the red-
dashed-dotted curves show the contribution from the $\Sigma^{*}(1/2^{-})$
state, the green-dotted curves show the contribution from the $\Sigma(1385)$
state, and the black-solid curves show the results obtained with the total
amplitude. The Belle data are taken from Ref. [34].

Figure 9: The $\pi^{+}\Lambda$ invariant mass distribution of the
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay for (a) Fit A: without the
contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the
contribution from the $\Sigma^{*}(1/2^{-})$. The curves have the same meaning
as those of Fig. 8.

Figure 10: The $\pi^{+}\eta$ invariant mass distribution of the
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay for (a) Fit A: without the
contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the
contribution from the $\Sigma^{*}(1/2^{-})$. The curves have the same meaning
as those of Fig. 8.

Figure 11: The $\pi^{+}\eta$ invariant mass distribution of the
$\Lambda_{c}\to\eta\pi^{+}\Lambda$ decay with the cuts of $M_{\pi\Lambda}\geq
1450$ MeV and $M_{\eta\Lambda}\geq 1760$ MeV for (a) Fit A: without the
contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the
contribution from the $\Sigma^{*}(1/2^{-})$. The curves have the same meaning
as those of Fig. 8.

Figure 12: The decay angular distributions of the
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay for (a) Fit A: without the
contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the
contribution from the $\Sigma^{*}(1/2^{-})$. The curves have the same meaning
as those of Fig. 8.

Figure 13: The decay angular distributions of the
$\Lambda_{c}\to\eta\pi^{+}\Lambda$ decay restricted to $M_{\pi\Lambda}\geq
1450$ MeV and $M_{\eta\Lambda}\geq 1760$ MeV. (a) without resonance
$\Sigma(1/2^{-})$, (b) with resonance $\Sigma(1/2^{-})$.

Figure 14: The Dalitz plot of “$\pi^{+}\Lambda$” vs. “$\pi^{+}\eta$” in the $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay for (a) Fit A: without the contribution from the $\Sigma^{*}(1/2^{-})$, and (b) Fit B: with the contribution from the $\Sigma^{*}(1/2^{-})$. Table 1: The fitted parameter values in this work. parameters | without $\Sigma^{*}(1/2^{-})$ | with $\Sigma^{*}(1/2^{-})$
---|---|---
$V_{p}$ | $3.662\pm 0.034$ | $3.535\pm 0.031$
$V^{\Sigma(1385)}$ | $1.053\pm 0.007$ | $0.804\pm 0.014$
$a_{K\Xi}$ | $-2.655\pm 0.007$ | $-2.677\pm 0.007$
$\phi$ | $(0.162\pm 0.009)\pi$ | $(0.120\pm 0.013)\pi$
$\phi^{\prime}$ | $(1.091\pm 0.021)\pi$ | $(0.528\pm 0.034)\pi$
$\phi^{\prime\prime}$ |  | $(-6.46\pm 1.52)\times 10^{-2}\pi$
$V^{\Sigma(1/2^{-})}$ |  | $1.506\pm 0.033$
$\chi^{2}/d.o.f.$ | $3.04$ | $1.83$

In our model, we have seven free parameters, which are, (1) the strength
$V_{p}$ related to the meson-baryon interaction of Fig. 1; (2) the subtraction
constant $a_{K\Xi}$ for the $K\Xi$ channel appearing in Eq. (10); (3) the
strength $V^{\Sigma(1385)}$ for the contribution from $\Sigma(1385)$; (4) the
strength $V^{\Sigma(1/2^{-})}$ for the contribution from
$\Sigma^{*}(1/2^{-})$; (5) three phase angles $\phi$, $\phi^{\prime}$,
$\phi^{\prime\prime}$ for the interference between different contributions
appearing in Eq. (20).

The Belle Collaboration has reported the $\eta\Lambda$ and $\pi^{+}\Lambda$
invariant mass distributions of the process
$\Lambda_{c}^{+}\to\eta\pi^{+}\Lambda$ [34]. In this work, we perform two
kinds of fits to the Belle measurements, (a) without the contribution from the
$\Sigma(1/2^{-})$ (Fit A), (b) with the contribution from the
$\Sigma(1/2^{-})$ (Fit B). In Fit A, we have five free parameters since the
$\phi^{\prime\prime}$ and $V^{\Sigma(1/2^{-})}$ are related to the
$\Sigma^{*}(1/2^{-})$. The fitted parameters are tabulated in Table 1. One can
see that our models of either Fit A or Fit B could describe the present Belle
measurements. However, the $\chi^{2}/d.o.f.$ of Fit B is 1.83, better than
that (3.04) of Fit A, which implies that considering the contribution from the
$\Sigma^{*}(1/2^{-})$ could significantly improve the description of the Belle
measurements. On the other hand, both fitted values of $a_{K\Xi}$ are close to
those obtained in Refs. [14, 35]. In Fig. 7, we have presented the modulus
squared of the transition amplitude $|T_{K\Xi\to\eta\Lambda}|^{2}$ with
$a_{K\Xi}=-2.655$ and $a_{K\Xi}=-2.677$, where one can find that both pole
positions are around 1680 MeV, consistent with the RPP [11].

With the fitted parameters, we have calculated the invariant mass
distributions of $\eta\Lambda$ and $\pi^{+}\Lambda$, as shown in Fig. 8 and
Fig. 9, respectively. The left and right panels correspond to the results of
Fit A and Fit B, respectively. It should be stressed that two peaks appear
around 1910 MeV and 2090 MeV in the $\eta\Lambda$ invariant mass distribution,
which is due to the contribution from the intermediate resonance
$\Sigma(1385)$, and should not be associated with any resonance. Since the
contribution from $\Sigma(1385)$ will be weakened when the
$\Sigma^{*}(1/2^{-})$ is involved, and the lineshape of the
$\Sigma^{*}(1/2^{-})$ is smooth, it will lead to different strengths of two
peak structures for Fit A and Fit B, which could serve as one probe to
demonstrate the existence of $\Sigma^{*}(1/2^{-})$.

Next, we show the $\pi^{+}\eta$ invariant mass distributions in Figs. 10(a)
and 10(b), which correspond to the results of Fit A and Fit B, respectively.
One could find a clear cusp structure around $M_{\eta\pi^{+}}=980$ MeV, which
could be associated with the resonance $a_{0}(980)$. Indeed, the cusp
structure of the scalar meson $a_{0}(980)$ has been observed in many
processes, such as $D^{+}\to K_{s}\pi^{+}\eta$ [47, 48],
$\chi_{c1}\to\eta\pi^{+}\pi^{-}$ [49, 50]. In addition, two peaks appear
around the $\eta\pi^{+}$ threshold and $M_{\eta\pi^{+}}=1100$ MeV mainly due
to the contribution from the $\Sigma(1385)$. Although the lineshape of the
$\eta\pi^{+}$ invariant mass distributions obtained in Fit A and Fit B are
different, it is difficult to distinguish these two cases considering the
unavoidable uncertainties of future experiments. The main reason is that the
contributions from the $\Sigma(1385)$, $\Sigma^{*}(1/2^{-})$, and
$\Lambda{(1670)}$ is strong to bury the pure contribution of $a_{0}(980)$.

Thus, we take the cuts of $M_{\pi\Lambda}\geq 1450$ MeV and
$M_{\eta\Lambda}\geq 1760$ MeV to eliminate the contributions from the
$\Sigma(1385)$, $\Sigma^{*}(1/2^{-})$, and $\Lambda(1670)$, and show the
results in Fig. 11, where the lineshapes obtained in Fit A and Fit B are
significantly different, which could be used to test the existence of
$\Sigma^{*}(1/2^{-})$. Such large disperse is from the interference between
the $a_{0}(980)$ and $\Sigma^{*}$ resonances, where in Fit A the
$\Sigma(1385)$ is too narrow to affect the region of $M_{\pi\Lambda}\geq 1450$
MeV, while in Fit B the $\Sigma^{*}(1/2^{-})$ still provides important
contribution to $M_{\pi\Lambda}\geq 1450$ MeV because of its large width.

With Eq. (25), we have calculated the angular distributions of the
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay with the fitted parameters of Fit
A and Fit B, as shown in Figs. 12(a) and 12(b), respectively. It is expected
that there will be more events when $\pi^{+}$ and $\eta$ are parallel or anti-
parallel, mainly due to the contributions from the $\Sigma(1385)$ and
$\Lambda(1670)$. By performing the cuts of $M_{\pi\Lambda}\geq 1450$ MeV and
$M_{\eta\Lambda}\geq 1760$ MeV to eliminate the contributions from the
$\Sigma(1385)$, $\Sigma^{*}(1/2^{-})$, and $\Lambda(1670)$, we present the
results in Figs. 13(a) and 13(b) for the cases of Fit A and Fit B,
respectively. A clear peak appears around ${\rm cos}\theta=-0.1$ in both
cases. However, more events are expected in the region of negative ${\rm
cos}\theta$ for the case of Fit A, while for the case of Fit B, more events
are expected in the region of positive ${\rm cos}\theta$, which could also be
used to test the existence of the $\Sigma^{*}(1/2^{-})$. Again, it is due to
the interference between the $a_{0}(980)$ and $\Sigma^{*}(1/2^{-})$ is large
and therefore reduces the total amplitude at $\theta\sim\pi$ and $0$,
respectively.

Finally, we present the Dalitz plots of “$\pi^{+}\Lambda$” vs. “$\eta\Lambda$”
with the fitted parameters of Fit A and Fit B in Figs. 14(a) and 14(b),
respectively. In both cases, one can find the clear signal of the
$\Sigma(1385)$ around $M^{2}_{\pi\Lambda}=1.8\sim 2.0$ GeV2, and a green band
around $M^{2}_{\eta\Lambda}=2.8$ GeV2, which should be associated with the
$\Lambda(1670)$ state. In addition, one can easily find a thin band
corresponding to the $a_{0}(980)$. It should be pointed out that more events
are expected in the region of $M^{2}_{\Lambda\pi^{+}}<1.85$ GeV2 and
$M^{2}_{\eta\Lambda}<4$ GeV2 when the contribution from the
$\Sigma^{*}(1/2^{-})$ is taken into account, as shown in Fig. 14(b). It is in
good agreement with the Belle measurements. Thus, one could conclude that the
existence of the $\Sigma^{*}(1/2^{-})$ is favored by the Belle measurements of
the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$. Unfortunately, we can not
perform a fit for such Dalitz plots but only a rough comparison. From the
above discussion, we expect that the invariant mass spectrum of $\eta\pi^{+}$
and angular distributions, especially after the cuts, can help us establish
the existence of the $\Sigma^{*}(1/2^{-})$ particle around 1400 MeV.

##  IV  Conclusions

Recently, the Belle Collaboration measured the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ and reported the $\eta\Lambda$ and
$\Lambda\pi$ invariant mass distributions and the Dalitz plot, which show the
signals of the resonances $\Lambda(1670)$, $a_{0}(980)$, and $\Sigma(1385)$.
In this work, we have investigated this process by considering the
contributions from the $S$-wave $\eta\Lambda$ and $\eta\pi$ final-state
interactions, which dynamically generate the $\Lambda(1670)$ and $a_{0}(980)$,
respectively. We have also considered contributions from the intermediate
resonances $\Sigma(1385)$ with $J^{P}=3/2^{+}$ and the predicted state
$\Sigma^{*}(1/2^{-})$.

We have performed two kinds of fits to the Belle measurements of the
$\eta\Lambda$ and $\pi\Lambda$ invariant mass distributions, without (Fit A)
and with (Fit B) including the contribution of the $\Sigma^{*}(1/2^{-})$.
Although our results obtained in both cases could describe the Belle
measurements of the $\eta\Lambda$ and $\pi\Lambda$ invariant mass
distributions, the $\chi^{2}/d.o.f.$ of Fit B is 1.83, better than that (3.04)
of Fit A, which implies that considering the contribution from the
$\Sigma^{*}(1/2^{-})$ could significantly improve the descriptions of the
Belle measurements. But, to be fair, the peaks in the invariant mass of the
$\eta\Lambda$ and $\pi\Lambda$ can be well explained by both fits.

Then, we predicted the $\eta\pi^{+}$ invariant mass distributions and found
one clear peak structure around $980$ MeV, which should be associated with the
scalar meson $a_{0}(980)$. By performing the cuts of $M_{\pi^{+}\Lambda}\geq
1450$ MeV and $M_{\eta\Lambda}\geq 1760$ MeV to eliminate the contributions
from the $\Sigma(1385)$, $\Sigma^{*}(1/2^{-})$, and $\Lambda(1670)$, the line
shapes of the $\eta\pi^{+}$ invariant mass distributions are significantly
different when considering the contribution from the $\Sigma^{*}(1/2^{-})$.
Furthermore, we have also predicted the angular distribution for this process
and found that there will be more events when $\pi^{+}$ and $\eta$ are
parallel or anti-parallel, which is mainly due to the contributions from the
$\Sigma(1385)$ and $\Lambda(1670)$. By performing the cuts of
$M_{\pi^{+}\Lambda}\geq 1450$ MeV and $M_{\eta\Lambda}\geq 1760$ MeV to
eliminate the contributions from the $\Sigma(1385)$, $\Sigma^{*}(1/2^{-})$,
and $\Lambda(1670)$, one finds more events in the region of positive ${\rm
cos}\theta$. All of these differences stem from the interference between the
$a_{0}(980)$ and $\Sigma^{*}(1/2^{-})$; thus, it provides a unique place to
identify the existence of $\Sigma^{*}(1/2^{-})$ around 1.4 GeV.

With the fitted parameters in both cases, we have calculated the Dalitz plots
and found clear signals of the $\Lambda(1670)$, $\Sigma(1385)$, and
$a_{0}(980)$. It should be pointed out that more events are expected in the
region of $M^{2}_{\Lambda\pi^{+}}<1.85$ GeV2 and $M^{2}_{\eta\Lambda}<4$ GeV2
when the contribution from the $\Sigma^{*}(1/2^{-})$ is taken into account,
which agree well with the Belle measurements. Thus, one could conclude that
the existence of the $\Sigma^{*}(1/2^{-})$ is favored by the Belle
measurements of the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$, and our
predictions could be helpful for future experimental analysis of this process.

## Acknowledgements

LSG and JJX are partly supported by the National Key R&D Program of China
under Grant No. 2023YFA1606700. This work is supported by the Natural Science
Foundation of Henan under Grant No. 232300421140 and No. 222300420554, the
National Natural Science Foundation of China under Grant No. 12192263,
12205075, 12175239, 12221005, 12075288, 12361141819, and also by the Project
of Youth Backbone Teachers of Colleges and Universities of Henan Province
(2020GGJS017), and also by the Open Project of Guangxi Key Laboratory of
Nuclear Physics and Nuclear Technology, No. NLK2021-08, and also by the
National Key Research and Development Program of China under Contracts
2020YFA0406400, and also by the Chinese Academy of Sciences under Grant No.
YSBR-101, and also by the Youth Innovation Promotion Association CAS.

## References

  * [1] S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003) doi:10.1103/PhysRevLett.91.262001 [arXiv:hep-ex/0309032 [hep-ex]].
  * [2] H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 80, no.7, 076201 (2017) doi:10.1088/1361-6633/aa6420 [arXiv:1609.08928 [hep-ph]].
  * [3] H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86 (2023) no.2, 026201 doi:10.1088/1361-6633/aca3b6 [arXiv:2204.02649 [hep-ph]].
  * [4] F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90 (2018) no.1, 015004 [erratum: Rev. Mod. Phys. 94 (2022) no.2, 029901] doi:10.1103/RevModPhys.90.015004 [arXiv:1705.00141 [hep-ph]].
  * [5] E. Oset, W. H. Liang, M. Bayar, J. J. Xie, L. R. Dai, M. Albaladejo, M. Nielsen, T. Sekihara, F. Navarra and L. Roca, et al. Int. J. Mod. Phys. E 25 (2016), 1630001 doi:10.1142/S0218301316300010 [arXiv:1601.03972 [hep-ph]].
  * [6] M. Z. Liu, Y. W. Pan, Z. W. Liu, T. W. Wu, J. X. Lu and L. S. Geng, [arXiv:2404.06399 [hep-ph]].
  * [7] H. Y. Gao and B. Q. Ma, Mod. Phys. Lett. A 14 (1999), 2313-2319 doi:10.1142/S021773239900239X [arXiv:hep-ph/0305294 [hep-ph]].
  * [8] E. Oset and A. Ramos, Nucl. Phys. A 635 (1998), 99-120 doi:10.1016/S0375-9474(98)00170-5 [arXiv:nucl-th/9711022 [nucl-th]].
  * [9] D. Jido, J. A. Oller, E. Oset, A. Ramos and U. G. Meissner, Nucl. Phys. A 725 (2003), 181-200 doi:10.1016/S0375-9474(03)01598-7 [arXiv:nucl-th/0303062 [nucl-th]].
  * [10] T. Inoue, E. Oset and M. J. Vicente Vacas, Phys. Rev. C 65, 035204 (2002) doi:10.1103/PhysRevC.65.035204 [arXiv:hep-ph/0110333 [hep-ph]].
  * [11] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * [12] V. Crede and W. Roberts, Rept. Prog. Phys. 76, 076301 (2013) doi:10.1088/0034-4885/76/7/076301 [arXiv:1302.7299 [nucl-ex]].
  * [13] E. Klempt and J. M. Richard, Rev. Mod. Phys. 82, 1095-1153 (2010) doi:10.1103/RevModPhys.82.1095 [arXiv:0901.2055 [hep-ph]].
  * [14] E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527, 99-105 (2002) [erratum: Phys. Lett. B 530, 260-260 (2002)] doi:10.1016/S0370-2693(01)01523-4 [arXiv:nucl-th/0109006 [nucl-th]].
  * [15] J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263-272 (2001) doi:10.1016/S0370-2693(01)00078-8 [arXiv:hep-ph/0011146 [hep-ph]].
  * [16] K. P. Khemchandani, A. Martínez Torres and J. A. Oller, Phys. Rev. C 100, no.1, 015208 (2019) doi:10.1103/PhysRevC.100.015208 [arXiv:1810.09990 [hep-ph]].
  * [17] Y. Kamiya, K. Miyahara, S. Ohnishi, Y. Ikeda, T. Hyodo, E. Oset and W. Weise, Nucl. Phys. A 954, 41-57 (2016) doi:10.1016/j.nuclphysa.2016.04.013 [arXiv:1602.08852 [hep-ph]].
  * [18] J. A. Oller, Eur. Phys. J. A 28, 63-82 (2006) doi:10.1140/epja/i2006-10011-3 [arXiv:hep-ph/0603134 [hep-ph]].
  * [19] C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D 67, 076009 (2003) doi:10.1103/PhysRevD.67.076009 [arXiv:hep-ph/0210311 [hep-ph]].
  * [20] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A 700, 193-308 (2002) doi:10.1016/S0375-9474(01)01312-4 [arXiv:nucl-th/0105042 [nucl-th]].
  * [21] L. Roca and E. Oset, Phys. Rev. C 88, no.5, 055206 (2013) doi:10.1103/PhysRevC.88.055206 [arXiv:1307.5752 [nucl-th]].
  * [22] K. Moriya et al. [CLAS], Phys. Rev. C 87, no.3, 035206 (2013) doi:10.1103/PhysRevC.87.035206 [arXiv:1301.5000 [nucl-ex]].
  * [23] J. J. Wu, S. Dulat and B. S. Zou, Phys. Rev. D 80, 017503 (2009) doi:10.1103/PhysRevD.80.017503 [arXiv:0906.3950 [hep-ph]].
  * [24] J. J. Wu, S. Dulat and B. S. Zou, Phys. Rev. C 81, 045210 (2010) doi:10.1103/PhysRevC.81.045210 [arXiv:0909.1380 [hep-ph]].
  * [25] P. Gao, J. J. Wu and B. S. Zou, Phys. Rev. C 81, 055203 (2010) doi:10.1103/PhysRevC.81.055203 [arXiv:1001.0805 [nucl-th]].
  * [26] J. J. Xie and L. S. Geng, Phys. Rev. D 95, no.7, 074024 (2017) doi:10.1103/PhysRevD.95.074024 [arXiv:1703.09502 [hep-ph]].
  * [27] L. J. Liu, E. Wang, J. J. Xie, K. L. Song and J. Y. Zhu, Phys. Rev. D 98, no.11, 114017 (2018) doi:10.1103/PhysRevD.98.114017 [arXiv:1712.07469 [hep-ph]].
  * [28] E. Wang, J. J. Xie and E. Oset, Phys. Lett. B 753, 526-532 (2016) doi:10.1016/j.physletb.2015.12.060 [arXiv:1509.03367 [hep-ph]].
  * [29] Y. H. Lyu, H. Zhang, N. C. Wei, B. C. Ke, E. Wang and J. J. Xie, Chin. Phys. C 47 (2023) no.5, 053108 [arXiv:2301.08615 [hep-ph]].
  * [30] S. H. Kim, K. P. Khemchandani, A. Martinez Torres, S. i. Nam and A. Hosaka, Phys. Rev. D 103 (2021) no.11, 114017 doi:10.1103/PhysRevD.103.114017 [arXiv:2101.08668 [hep-ph]].
  * [31] J. X. Lu, L. S. Geng, M. Doering and M. Mai, Phys. Rev. Lett. 130 (2023) no.7, 071902 doi:10.1103/PhysRevLett.130.071902 [arXiv:2209.02471 [hep-ph]].
  * [32] R. Ammar et al. [CLEO], Phys. Rev. Lett. 74 (1995), 3534-3537 doi:10.1103/PhysRevLett.74.3534
  * [33] M. Ablikim et al. [BESIII], Phys. Rev. D 99 (2019) no.3, 032010 doi:10.1103/PhysRevD.99.032010 [arXiv:1812.10731 [hep-ex]].
  * [34] J. Y. Lee et al. [Belle], Phys. Rev. D 103, no.5, 052005 (2021) doi:10.1103/PhysRevD.103.052005 [arXiv:2008.11575 [hep-ex]].
  * [35] G. Y. Wang, N. C. Wei, H. M. Yang, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 106, no.5, 056001 (2022) doi:10.1103/PhysRevD.106.056001 [arXiv:2206.01425 [hep-ph]].
  * [36] A. Bramon, A. Grau and G. Pancheri, Phys. Lett. B 283 (1992), 416-420 doi:10.1016/0370-2693(92)90041-2
  * [37] W. T. Lyu, Y. H. Lyu, M. Y. Duan, D. M. Li, D. Y. Chen and E. Wang, Phys. Rev. D 109 (2024) no.1, 014008 doi:10.1103/PhysRevD.109.014008 [arXiv:2306.16101 [hep-ph]].
  * [38] W. T. Lyu, Y. H. Lyu, M. Y. Duan, G. Y. Wang, D. Y. Chen and E. Wang, [arXiv:2310.11139 [hep-ph]].
  * [39] R. P. Pavao, W. H. Liang, J. Nieves and E. Oset, Eur. Phys. J. C 77, no.4, 265 (2017) doi:10.1140/epjc/s10052-017-4836-3 [arXiv:1701.06914 [hep-ph]].
  * [40] K. Miyahara, T. Hyodo, M. Oka, J. Nieves and E. Oset, Phys. Rev. C 95, no.3, 035212 (2017) doi:10.1103/PhysRevC.95.035212 [arXiv:1609.00895 [nucl-th]].
  * [41] L. Roca, M. Mai, E. Oset and U. G. Meißner, Eur. Phys. J. C 75, no.5, 218 (2015) doi:10.1140/epjc/s10052-015-3438-1 [arXiv:1503.02936 [hep-ph]].
  * [42] J. J. Xie and L. S. Geng, Eur. Phys. J. C 76, no.9, 496 (2016) doi:10.1140/epjc/s10052-016-4342-z [arXiv:1604.02756 [nucl-th]].
  * [43] J. J. Xie, L. R. Dai and E. Oset, Phys. Lett. B 742 (2015), 363-369 doi:10.1016/j.physletb.2015.02.006 [arXiv:1409.0401 [hep-ph]].
  * [44] J. Y. Wang, M. Y. Duan, G. Y. Wang, D. M. Li, L. J. Liu and E. Wang, Phys. Lett. B 821 (2021), 136617 doi:10.1016/j.physletb.2021.136617 [arXiv:2105.04907 [hep-ph]].
  * [45] X. C. Feng, L. L. Wei, M. Y. Duan, E. Wang and D. M. Li, Phys. Lett. B 846 (2023), 138185 doi:10.1016/j.physletb.2023.138185 [arXiv:2009.08600 [hep-ph]].
  * [46] E. Wang, H. X. Chen, L. S. Geng, D. M. Li and E. Oset, Phys. Rev. D 93, no.9, 094001 (2016) doi:10.1103/PhysRevD.93.094001 [arXiv:1512.01959 [hep-ph]].
  * [47] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 132 (2024) no.13, 131903 doi:10.1103/PhysRevLett.132.131903 [arXiv:2309.05760 [hep-ex]].
  * [48] M. Y. Duan, J. Y. Wang, G. Y. Wang, E. Wang and D. M. Li, Eur. Phys. J. C 80 (2020) no.11, 1041 doi:10.1140/epjc/s10052-020-08630-3 [arXiv:2008.10139 [hep-ph]].
  * [49] W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C 76 (2016) no.12, 700 doi:10.1140/epjc/s10052-016-4563-1 [arXiv:1609.03864 [hep-ph]].
  * [50] M. Ablikim et al. [BESIII], Phys. Rev. D 95 (2017) no.3, 032002 doi:10.1103/PhysRevD.95.032002 [arXiv:1610.02479 [hep-ex]].


# Using propensity scores for racial disparities analysis

Fan Li<EMAIL_ADDRESS>
Department of Statistical Science
Duke University
Durham, North Carolina, 27708, USA Fan Li<EMAIL_ADDRESS>
Department of Biostatistics
Yale University
New Haven, Connecticut, 06510, USA

###### Abstract

Propensity score plays a central role in causal inference, but its use is not
limited to causal comparisons. As a covariate balancing tool, propensity score
can be used for controlled descriptive comparisons between groups whose
memberships are not manipulable. A prominent example is racial disparities in
health care. However, conceptual confusion and hesitation persists for using
propensity score in racial disparities studies. In this commentary, we argue
that propensity score, possibly combined with other methods, is an effective
tool for racial disparities analysis. We describe relevant estimands, target
population, and assumptions. In particular, we clarify that a controlled
descriptive comparisons require weaker assumptions than a causal comparison.
We discuss three common propensity score weighting strategies: overlap
weighting, inverse probability weighting and average treatment effect for
treated weighting. We further describe how to combine weighting with the rank-
and-replace adjustment method to produce racial disparity estimates concordant
to the Institute of Medicine’s definition. The method is illustrated by a re-
analysis of the Medical Expenditure Panel Survey data.

Keywords: covariate balance; propensity score weighting; racial disparities;
target population; controlled descriptive comparison

##  1 Introduction

Since first introduced in the landmark Rosenbaum and Rubin (1983) paper,
propensity score has become a central concept in the potential outcome
framework for causal inference. Following the dictum “ _no causation without
manipulation_ ” (Holland, 1986), a “cause” refers to a _treatment_ or
_intervention_ that is at least hypothetically manipulable. Hence for many
researchers, the concept of propensity score is unequivocally tied with causal
inference for manipulable interventions. However, a key property of the
propensity score, the balancing property, does not involve potential outcomes
nor imply causal interpretation. In fact, propensity scores have already been
used in several non-causal contexts, e.g. generalizability (Stuart et al.,
2011), constructing external controls in clinical trials from real world data
(Lim et al., 2018), covariate adjustment in randomized trials (Zeng et al.,
2021). But a prominent exception is racial disparity studies, where the
comparison is between different racial groups. Because race is not
manipulable, racial disparity investigations are inherently not causal
(Holland, 2003; Zaslavsky and Ayanian, 2005), and thus there is often
hesitation among health service researchers to use propensity scores for
studying racial disparities.

In this commentary, we aim to elucidate the basis and conditions for using
propensity score for non-causal comparisons. The central message is that
propensity score, as is defined in the broad sense, is a one-dimensional
summary of the collection of pre-treatment covariates, and can be used as a
numerical tool for balancing covariates between different groups, regardless
whether the group assignment is (hypothetically) manipulable. Given the
persistent misconception about propensity scores in racial disparities
analysis, below we focus on the context of racial disparities in health care
utilization. We highlight the importance of specifying target population _a
priori_ and separating operational properties and contextual interpretation in
propensity score analyses. We stress that our perspective is purely
descriptive, and thus shall not be confused with the literature that addresses
causal interpretation of racial comparisons, as in VanderWeele and Robinson
(2014a, b) and the references therein.

##  2 Controlled descriptive comparisons: estimands, target population and
assumptions

Let $Z$ denote the group membership variable, which is assumed to be binary
for simplicity (e.g. White versus non-White), $X$ denote a set of measured
covariates, and $Y$ the observed health care outcome of interest. Because the
goal is descriptive rather than causal comparisons, “assignment” here refers
to a nonmanipulable state defining membership in one of two groups or
populations, and the objective is a controlled comparison of the observed
outcomes between the groups after adjusting for the differences in a set of
pre-treatment covariates. Because confounding is a term closely tied to causal
inference, we choose the term _controlled_ to emphasize the descriptive nature
of the comparison. The propensity score $e(x)=\Pr(Z=1\mid X=x)$ is the
conditional probability of being in group $Z=1$ given covariates $X$.

Following Li et al. (2013), we define the conditional average controlled
difference (ACD) given covariate value $x$ as,

$\tau(x)\equiv E(Y\mid Z=1,X=x)-E(Y\mid Z=0,X=x).$ (1)

Then we can define the average controlled difference on a scientifically
meaningful target population by averaging the conditional ACD over that
population, which can be represented as a weighted average controlled
difference (WACD). Specifically, assume the observed sample is drawn from a
population with probability density of covariates $f(X)$. Let $g(X)$ denote
the covariate density of a pre-specified target population, which may be
different from $f(X)$. We call the ratio $h(X)=g(X)/f(X)$ the _tilting
function_ , which re-weights the distribution of the observed sample to
represent the target population. Then we can represent the average controlled
difference on the target population $g$ by a WACD estimand:

$\tau_{h}=E_{g}[\tau(X)]=\frac{E[h(X)\tau(X)]}{E[h(X)]}.$ (2)

In plain language, $\tau_{h}$ refers to the average net difference in the
health care outcome $Y$ between two groups with their covariate distributions
adjusted to be the same as in the target population $g$. Different tilting
functions lead to different target population, as illustrated in the following
three examples. For notational purposes, we define $\mathbb{S}_{0}$ and
$\mathbb{S}_{1}$ as the units in the $Z=0$ group and the $Z=1$ group,
respectively.

  * •

When $h(X)\propto 1$, $\tau_{h}$ represents the average difference in the
outcome $Y$ once we force the distribution of $X$ in each of $\mathbb{S}_{0}$,
$\mathbb{S}_{1}$ to be identical to that in the union
$\mathbb{S}_{0}\cup\mathbb{S}_{1}$, which is the target population. In the
racial disparity context, this target population would be the overall
population combining the two racial groups under comparison with all
covariates being balanced.

  * •

When $h(X)\propto e(X)$, $\tau_{h}$ represents the average difference in the
outcome $Y$ once we force the distribution of $X$ in group $\mathbb{S}_{0}$ to
be identical to that in group $\mathbb{S}_{1}$. The same argument applies to
$h(X)\propto\\{1-e(X)\\}$ if we flip the definition of $Z=1$ and $Z=0$. So the
target population is group $\mathbb{S}_{1}$ or $\mathbb{S}_{0}$. In the racial
disparity context, it would be a population with the same covariate
distribution as one pre-specified racial group.

  * •

When $h(X)\propto e(X)\\{1-e(X)\\}$, $\tau_{h}$ represents the average
difference in the outcome $Y$ once we force the distribution of $X$ in each of
$\mathbb{S}_{0}$, $\mathbb{S}_{1}$ to be identical to that in the
subpopulation that has the largest tendency to belong to both $\mathbb{S}_{0}$
and $\mathbb{S}_{1}$, a concept similar to equipoise in clinical evaluations.
We call this target population the overlap population (Li et al., 2018, 2019;
Cheng et al., 2022), which is akin to an intersection—rather than a
union—between $\mathbb{S}_{0}$ and $\mathbb{S}_{1}$. In the racial disparity
context, this target population would be the subpopulation with the most
similar covariate distribution between the two racial groups.

We stress a key but under-appreciated distinction between causal and
controlled descriptive comparisons in terms of necessary assumptions, namely,
the latter requires weaker assumptions than the former. Causal comparisons
typically require (A1) the Stable Unit Treatment Value Assumption (SUTVA),
which states there is no interference between units and no different versions
of the treatment; (A2) the unconfoundedness assumption, which states there is
no unmeasured confounder; and (A3) the overlap or positivity assumption, which
states that each unit has non-zero probability of being in either group. SUTVA
underpins the existence of two potential outcomes for each unit;
uconfoundedness connects potential outcomes to the observed outcomes and
enables interpreting the difference in the observed outcomes as a causal
effect. However, a controlled descriptive comparison merely needs to adjust
for the difference in the covariates between two groups rather than offers a
causal interpretation of the difference in the observed outcome. Therefore, it
does not involve the concept of potential outcomes and consequently does not
require SUTVA or unconfoundedness. On the other hand, the overlap assumption
is necessary for both causal and controlled descriptive comparisons. This is
first for an operational reason because propensities close to 0 and
1—equivalently lack of overlap—lead to large variance in estimation. Moreover,
overlap is needed in causal comparisons for an additional conceptual reason to
define causal effects: if a unit has zero probability to be assigned to one
group, then we cannot conceive its potential outcome corresponding to that
group assignment.

##  3 Propensity score methods for estimation

We now outline two estimation methods of the WACD: matching and weighting. In
matching, one chooses an algorithm finding pairs of units in two groups with
similar covariates according to some distance metric (e.g. the propensity
score), and then calculates the difference in the average observed outcome
between the groups in the matched sample (Rubin, 2006). The overlap assumption
is in effect achieved by dropping the unmatched units. Matching is a bottom-up
method in the sense that it starts from local balance in the observed sample
rather than global balance in a pre-defined target population. The target
population is defined only implicitly through the matching algorithm. For
example, when the matching algorithm is designed to find matches for (nearly)
all units in the whole sample, e.g. the full matching method (Rosenbaum, 1991;
Hansen, 2004), the target population is the overall population. When the
algorithm is designed to find matches for (nearly) all units from one specific
group (e.g. the treated group in the causal context or a racial group in the
disparities context), the target population is the population in that group,
corresponding to $\tau_{h}$ with $h(X)=e(X)$. When the matching algorithm is
designed to find matches for a subset of marginal units, that is, units who
might or might not belong to a specific group, e.g. the optimal matching
method (Rosenbaum, 2012), the target population is the overlap population,
corresponding to $\tau_{h}$ with $h(X)=e(X)\\{1-e(X)\\}$.

Weighting methods assign a weight to each unit and then calculate the weighted
difference in outcomes between the comparison groups. Weighting is a top-down
method in the sense that it starts from the global balance in a target
population rather than local balance in pairs in the sample. The target
population is explicitly pre-specified and determines the corresponding
weighting scheme (as elaborated below). Unlike matching, there is no automatic
procedural guarantee of the overlap assumption in weighting, which may cause
inflated variances in some weighting schemes such as the inverse probability
weighting. Below we will focus on the weighting method because it more
directly connects to the definition of the WACD estimands.

For any pre-specified target population $g(X)$ and equivalently the tilting
function $h(X)$, we define the corresponding balancing weights (Li et al.,
2018):

$\left\\{\begin{array}[]{ll}w_{1}(X)\propto{h(X)}/{e(X)},&\mbox{for }Z=1\\\
w_{0}(X)\propto{h(X)}/{(1-e(X))},&\mbox{for }Z=0,\end{array}\right.$

It is straightforward to show that the WACD estimand (2) can be represented as
the weighted difference in the mean outcome between the two groups

$\displaystyle\tau_{h}=\frac{\mathbb{E}[YZw_{1}(X)]}{\mathbb{E}[Zw_{1}(X)]}-\frac{\mathbb{E}[Y(1-Z)w_{0}(X)]}{\mathbb{E}[(1-Z)w_{0}(X)]},$

without invoking SUTVA or the unconfoundedness assumption. To estimate WACD,
one can then consider the following Hajék estimator

$\displaystyle\widehat{\tau}_{h}=\frac{\sum_{i=1}^{N}Y_{i}Z_{i}w_{1}(X_{i})}{\sum_{i=1}^{N}Z_{i}w_{1}(X_{i})}-\frac{\sum_{i=1}^{N}Y_{i}(1-Z_{i})w_{0}(X_{i})}{\sum_{i=1}^{N}(1-Z_{i})w_{0}(X_{i})},$
(3)

where the weights $\\{w_{1}(X_{i}),w_{0}(X_{i})\\}$ are based on estimated
propensity scores $\widehat{e}(X_{i})$, e.g. via a logistic regression.

Here we list the balancing weights corresponding to previously discussed three
target populations. When $h(X_{i})\propto 1$,
$\\{w_{1}(X_{i})=1/e(X_{i}),w_{0}(X_{i})=1/\\{1-e(X_{i})\\}\\}$ is the inverse
probability weight (IPW). Conceptually, IPW balances the covariates toward the
overall population represented by the sample. Operationally, IPW over-weight
units who have large probability to being in the opposite group (i.e. with
propensity scores close to 0 for units in $\mathbb{S}_{1}$ or close to 1 for
units in $\mathbb{S}_{0}$). In other words, these units’ characteristics are
the least similar to the opposite group. Such a feature is not desirable in
the context of racial disparities because arguably we want to over-weight the
units who are the most similar between groups (i.e., with propensity scores
close to 0.5). This is exactly what overlap weight (OW),
$\\{w_{1}(X_{i})=\\{1-e(X_{i})\\},w_{0}(X_{i})=e(X_{i})\\}$ (corresponding to
$h(X_{i})\propto e(X_{i})\\{1-e(X_{i})\\}$), is designed to achieve.
Conceptually, OW emphasizes a naturally comparable subpopulation with similar
health status, namely patients whose race category, conditional on their
health conditions and clinical need, are indistinguishable. Operationally, Li
et al. (2018) showed that the overlap weights lead to the smallest asymptotic
variance of $\hat{\tau}_{h}$ among balancing weights. Moreover, when the
propensity score is estimated by a logistic regression, the resulting overlap
weights lead to exact mean balance of any covariate $X$ included in the
regression. Therefore, the overlap weights remove all imbalances in measured
covariates $X$ among the overlap population, rendering it particularly
suitable for controlled descriptive comparisons. When $h(X_{i})\propto
e(X_{i})$, $\\{w_{1}(X_{i})=1,w_{0}(X_{i})=e(X_{i})/\\{1-e(X_{i})\\}\\}$ is
the so-called ATT (average treatment effect for the treated) weight. The
operating characteristics of ATT weights are similar to those of IPW, but
limit to one group.

##  4 Application to racial disparities analysis in health care utilization

###  4.1 Propensity score weighting analysis

The _Unequal Treatment_ report from the Institute of Medicine (IOM) defines
health care disparity as the difference in treatment provided to social groups
that is not justified by health status or treatment preference of the patient
(IOM, 2003). To be concordant with the IOM definition, analysts need to adjust
for the health status variables across different racial groups in disparity
studies (Cook et al., 2012).

For illustration, we apply propensity score weighting to the 2009 Medical
Expenditure Panel Survey (MEPS) to study the White-Asian disparities in
medical expenditure. The analysis is implemented using the R package PSweight
(Zhou et al., 2022). Details on other White-minority such as White-Black and
White-Hispanic comparisons can be found in Cook et al. (2010) and Li and Li
(2019). The sample contains 9830 non-Hispanic White and 1446 Asian adults aged
at least 18 years. Here the propensity score is the probability of being in
the White group, estimated from a logistic propensity score model controlling
for the following health status variables (denoted by $X_{H}$): body mass
index, SF-12 physical and mental component summary, self-reported health
status, measurements of health conditions including diabetes, blood pressure,
asthma, MI, stroke, age, gender, and marital status. Figure 1 presents the
distribution of the estimated propensity scores (for being in the White group)
by each observed group. There is a substantial proportion of units,
particularly Asians, having propensities close to 1, suggesting a lack of
overlap. We applied IPW, ATT weighting and OW to balance the covariates. The
resulting weighted covariate balance, measured by the Absolute Standardized
Difference (ASD), is displayed in Figure 2 for each covariate. A rule of thumb
for covariate balance in the literature is ASD smaller than $0.1$ (Austin and
Stuart, 2015). Clearly, both IPW and ATT lead to insufficient balance in a
number of covariates with the largest ASD close to 1, whereas OW leads to
identically zero ASD in all covariates, effectively removing any difference in
the outcomes that is attributable to the health status variables in the
overlap population.

Figure 1: Distribution of the estimated propensity scores for being in the
White group.

The estimated disparities in total health expenditure based on different
weighting methods are presented in Table 1. The standard errors are estimated
using the closed-form sandwich variance estimator in Li et al. (2019), which
accounts for the uncertainty in estimating the propensity scores. Table 1
shows that Whites spent $2167 (95% CI (117, 4217)) , $2310 (95% CI (244,
4376)), $1227 (95% CI (796, 1658)) more on health care than Asians, using IPW,
ATT and OW, respectively. This shows that disparity estimates may be sensitive
to the choice of target population used for balancing covariates. The White-
Asian disparity estimates from IPW and ATT are likely subject to bias because
IPW and ATT fail to adequately balance the health status variables, as shown
in Figure 2. Besides, the lack of overlap leads to much inflated standard
errors in IPW and ATT, both of which are five times of that of OW.

Figure 2: Love plot of the absolute standardized difference for each covariate in the original and weighted data. The dotted vertical line indicates the threshold for balance at $0.1$. IPW: inverse probability weighting; overlap: overlap weighting; treated: ATT weighting. Table 1: White-Asian disparities in total health care expenditure (in dollars). The point estimates are obtained as average controlled differences (before and after propensity score weighting). IPW: inverse probability weighting; ATT: ATT weighting; OW: overlap weighting; IOM-c: IOM definition concordant. | Unweighted | IPW | IPW (IOM-c) | ATT | ATT (IOM-c) | OW | OW (IOM-c)
---|---|---|---|---|---|---|---
Estimate | $2764$ | $2167$ | $-933$ | $2310$ | $-1011$ | $1227$ | $1063$
SE | $225$ | $1046$ | $1963$ | $1054$ | $2046$ | $220$ | $238$

###  4.2 Combine propensity score weighting with rank-and-replace adjustment

The IOM definition of disparity includes racial differences in utilization
mediated through factors other than health status and preference, such as many
social factors (McGuire et al., 2006). This renders the analyses that only
adjust for health status characteristics as in Section 4.1 inadequate. A
number of methods have been developed to adjust for socioeconomic status (SES)
information in racial disparities studies in health services (e.g. McGuire et
al., 2006; Cook et al., 2009). In particular, McGuire et al. (2006) advocate
to distinguish between health status variables ($X_{H}$) and SES variables
($X_{S}$) in analysis. If the health status variables $X_{H}$ are correlated
with the SES variables $X_{S}$, propensity score weighting adjusting for
$X_{H}$ (as that in Section 4.1) may inadvertently alter the distributions of
$X_{S}$ and only provide an approximation to the IOM-defined disparity (Balsa
et al., 2007). McGuire et al. (2006) developed the rank-and-replace adjustment
method to undo the undesired weighting of $X_{S}$ due to its correlation with
$X_{H}$. Below we combine the rank-and-replace adjustment with our propensity
score analysis of MEPS as a further illustration. Here the SES variables
include poverty status, education, health insurance and geographical region.
We impose a log-linear model to model the total health care expenditure as a
function of $X_{H}$, $X_{S}$ and the racial group indicator

$\displaystyle\log(\mathbb{E}[Y_{i}|X_{H,i},X_{S,i},Z_{i}])=\gamma_{0}+\gamma_{1}Z_{i}+X_{H,i}^{T}\gamma_{H}+X_{S,i}^{T}\gamma_{S}.$
(4)

We take the fitted value of $X_{S,i}^{T}\gamma_{S}$ as the individual SES
predictive index, and first obtain the propensity score weighted (according to
each specific weighting scheme) rank of $X_{S,i}^{T}\gamma_{S}$ within each
race. To restore the original group-specific SES distributions, we then
replace $X_{S,i}^{T}\gamma_{S}$ for unit $i$ with $X_{S,j}^{T}\gamma_{S}$ for
unit $j$ such that the weighted rank of unit $i$ equals the unweighted rank of
unit $j$ within each racial group. With this adjustment, the weighted
distribution of the SES index in each group is approximately the same as the
original distribution of the SES index in that group. We then predict the
expenditure outcome for each individual based on model (4), after the rank-
and-replace adjustment, and estimate the WACD using this predicted expenditure
outcome. The resulting disparities estimates become more IOM-concordant in the
sense that they recapture the racial differences in SES even after propensity
score weighting of the health status variables. The final disparity estimates
are obtained by re-weighting (e.g. via OW or IPW) the predicted total health
care expenditure in model (4) after the rank-and-replace adjustment.

Table 1 provides the disparity estimates by combining propensity score
weighting with rank-and-replace adjustment (labeled as IOM-c); the standard
error estimates are obtained based on $1000$ bootstrap replicates.
Interestingly, these estimates are more sensitive to the choice of the target
population than the original weighting analysis. This is particularly
noticeable with IPW. Specifically, IPW suggests that Whites spent on average
$933 less than Asians when only adjusting for the difference in health status
(but not the SES). As implied from the lack of balance in Figure 1 and by
numerous simulation studies (Li et al., 2019; Li and Li, 2019), the IPW
disparities estimates are likely subject to bias. In comparison, the
disparities estimates under OW appear more stable. In particular, when we only
address differences in health status variables, OW suggests that Whites spent
on average $1063 more on health care than Asians, after using rank-and-replace
adjustment to restore differences due to SES variables.

Finally, we note that one limitation in implementing the IOM concordant
disparities analysis, even after the rank-and-replace adjustment, is that
there is no consensus on how to measure patient preferences. More substantive
guidance on this would benefit racial disparities studies in health care.

## Acknowledgement

The authors are grateful to Elizabeth Stuart for initiating this topic in a
conversation, to Alan Zaslavsky and Peng Ding for insightful discussions, to
Chao Cheng for computational assistance in the MEPS analysis in Section 4.

## References

  * Austin and Stuart (2015) Peter C Austin and Elizabeth A Stuart.  Moving towards best practice when using inverse probability of treatment weighting (iptw) using the propensity score to estimate causal treatment effects in observational studies.  _Statistics in Medicine_ , 34(28):3661–3679, 2015\.
  * Balsa et al. (2007) Ana I Balsa, Zhun Cao, and Thomas G McGuire.  Does managed health care reduce health care disparities between minorities and Whites?  _Journal of Health Economics_ , 27(1):781–807, 2007.
  * Cheng et al. (2022) Chao Cheng, Fan Li, Laine E Thomas, and Fan Li.  Addressing extreme propensity scores in estimating counterfactual survival functions via the overlap weights.  _American Journal of Epidemiology_ , 191(6):1140–1151, 2022.
  * Cook et al. (2009) Benjamin L. Cook, Thomas G Mcguire, Ellen Meara, and Alan M Zaslavsky.  Adjusting for health status in non-linear models of health care disparities.  _Health Services and Outcomes Research Methodology_ , 9(1):1–21, 2009.
  * Cook et al. (2010) Benjamin L. Cook, Thomas G Mcguire, Kari Lock, and Alan M Zaslavsky.  Comparing methods of racial and ethnic disparities measurement across different settings of mental health care.  _Health Services Research_ , 45(3):825–847, 2010\.
  * Cook et al. (2012) Benjamin L. Cook, Thomas G. McGuire, and Alan M. Zaslavsky.  Measuring racial/ethnic disparities in health care: Methods and practical issues.  _Health Services Research_ , 47(3):1232–1254, 2012.
  * Hansen (2004) Ben B Hansen.  Full matching in an observational study of coaching for the sat.  _Journal of the American Statistical Association_ , 99(467):609–618, 2004.
  * Holland (1986) Paul W Holland.  Statistics and causal inference.  _Journal of the American statistical Association_ , 81(396):945–960, 1986.
  * Holland (2003) Paul W Holland.  Causation and race.  _ETS Research Report Series_ , 2003(1):i–21, 2003\.
  * IOM (2003) IOM.  _Unequal Treatment: Confronting Racial and Ethnic Disparities in Health Care_.  The National Academies Press, Washington, DC, 2003.
  * Li and Li (2019) Fan Li and Fan Li.  Propensity score weighting for causal inference with multiple treatments.  _The Annals of Applied Statistics_ , 13(4):2389–2415, 2019.  doi: 10.1214/19-AOAS1282.
  * Li et al. (2013) Fan Li, Alan M Zaslavsky, and Mary Beth Landrum.  Propensity score weighting with multilevel data.  _Statistics in Medicine_ , 32(19):3373–3387, 2013\.
  * Li et al. (2018) Fan Li, Kari Lock Morgan, and Alan M. Zaslavsky.  Balancing Covariates via Propensity Score Weighting.  _Journal of the American Statistical Association_ , 113(521):390–400, 2018.  doi: 10.1080/01621459.2016.1260466.
  * Li et al. (2019) Fan Li, Laine E. Thomas, and Fan Li.  Addressing extreme propensity scores via the overlap weights.  _American Journal of Epidemiology_ , 1(188):250–257, 2019.
  * Lim et al. (2018) Jessica Lim, Rosalind Walley, Jiacheng Yuan, Jeen Liu, Abhishek Dabral, Nicky Best, Andrew Grieve, Lisa Hampson, Josephine Wolfram, Phil Woodward, et al.  Minimizing patient burden through the use of historical subject-level data in innovative confirmatory clinical trials: review of methods and opportunities.  _Therapeutic innovation & regulatory science_, 52(5):546–559, 2018.
  * McGuire et al. (2006) Thomas G. McGuire, Margarita Alegria, Benjamin L. Cook, Kenneth B. Wells, and Alan M. Zaslavsky.  Implementing the Institute of Medicine definition of disparities: An application to mental health care.  _Health Services Research_ , 41(5):1979–2005, 2006.
  * Rosenbaum (1991) Paul R Rosenbaum.  A characterization of optimal designs for observational studies.  _Journal of the Royal Statistical Society: Series B (Methodological)_ , 53(3):597–610, 1991.
  * Rosenbaum (2012) Paul R Rosenbaum.  Optimal matching of an optimally chosen subset in observational studies.  _Journal of Computational and Graphical Statistics_ , 21(1):57–71, 2012.
  * Rosenbaum and Rubin (1983) Paul R. Rosenbaum and Donald B. Rubin.  The central role of the propensity score in observational studies for causal effects.  _Biometrika_ , 70(1):41–55, 1983.  doi: 10.1093/biomet/70.1.41.
  * Rubin (2006) Donald B Rubin.  _Matched sampling for causal effects_.  Cambridge University Press, 2006.
  * Stuart et al. (2011) Elizabeth A Stuart, Stephen R Cole, Catherine P Bradshaw, and Philip J Leaf.  The use of propensity scores to assess the generalizability of results from randomized trials.  _Journal of the Royal Statistical Society: Series A (Statistics in Society)_ , 174(2):369–386, 2011.
  * VanderWeele and Robinson (2014a) Tyler J. VanderWeele and Whitney R. Robinson.  On the causal interpretation of race in regressions adjusting for confounding and mediating variables.  _Epidemiology_ , 25(4):473–484, 2014a.
  * VanderWeele and Robinson (2014b) Tyler J. VanderWeele and Whitney R. Robinson.  Rejoinder: How to reduce racial disparities?: Upon what to intervene?  _Epidemiology_ , 25(4):491–493, 2014b.
  * Zaslavsky and Ayanian (2005) Alan M Zaslavsky and John Z Ayanian.  Integrating research on racial and ethnic disparities in health care over place and time.  _Medical Care_ , 43(4):303–307, 2005.
  * Zeng et al. (2021) Shuxi Zeng, Fan Li, Rui Wang, and Fan Li.  Propensity score weighting for covariate adjustment in randomized clinical trials.  _Statistics in Medicine_ , 40(4):842–858, 2021\.
  * Zhou et al. (2022) Tianhui Zhou, Guangyu Tong, Fan Li, Laine E Thomas, and Fan Li.  Psweight: An R package for propensity score weighting analysis.  _The R Journal_ , 2022.


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Discovering Variable Binding Circuitry with Desiderata

Xander Davies * 1  Max Nadeau * 1  Nikhil Prakash * 2  Tamar Rott Shaham 3
David Bau 2

††footnotetext: *Equal contribution 1Harvard University 2Northeastern
University 3MIT CSAIL. Correspondence to: Xander Davies
<EMAIL_ADDRESS>Max Nadeau<EMAIL_ADDRESS>
Workshop on Challenges in Deployable Generative AI at International Conference
on Machine Learning (ICML), Honolulu, Hawaii, USA. 2023. Copyright 2023 by the
author(s).

###### Abstract

Recent work has shown that computation in language models may be human-
understandable, with successful efforts to localize and intervene on both
single-unit features and input-output circuits. Here, we introduce an approach
which extends causal mediation experiments to automatically identify model
components responsible for performing a specific subtask by solely specifying
a set of desiderata, or causal attributes of the model components executing
that subtask. As a proof of concept, we apply our method to automatically
discover shared variable binding circuitry in LLaMA-13B, which retrieves
variable values for multiple arithmetic tasks. Our method successfully
localizes variable binding to only 9 attention heads (of the 1.6k) and one MLP
in the final token’s residual stream.

##  1 Introduction

Deploying powerful generative AI systems requires confidence in the
reliability of their outputs, especially with respect to certain high stakes
behaviors like manipulation or truthfulness (Carroll et al., 2023; Perez et
al., 2022). The emerging field of mechanistic interpretability seeks to make
model computation human-understandable by explaining the function of
particular model components and locating groups of model components
responsible for performing certain language tasks. Indeed, recent work has
successfully identify, localize and intervene in model computation (Li et al.,
2022; Burns et al., 2022; Wang et al., 2022; Conmy et al., 2023).

Here, we introduce an automated approach which extends activation patching
(Meng et al., 2022b; Vig et al., 2020) to localize components within neural
networks (_e.g._ attention heads, MLP layers) that responsible for performing
a specific subtask of model computation. Our method allows for quickly and
automatically localizing computation, while only requiring to specify
desiderata, or causal attributes of the target computation. As a proof of
concept, we apply our method to automatically discover shared variable binding
circuitry in LLaMA-13B (Touvron et al., 2023), which retrieves variable values
for several arithmetic operations.

#### Contributions.

In this ongoing work, we:

  1. 1.

Describe a methodology for localizing computation by enumerating desiderata
and learning a binary mask by performing causal interventions (Section 3, Fig.
1).

  2. 2.

Present initial results in applying this methodology to localize shared
variable binding circuitry (Section 4, Fig. 2).

Figure 1: Localizing computation with desiderata. The figure depicts training
with a single (original, alternate, target) tuple within a desideratum. We
learn a mask $w$ that combines activations from an alternate sequence $a$ into
the computation of the model on the input of the original sequence $o$ such
that the output $y$ moves towards the target $t$.

##  2 Background

#### Circuit analysis.

A deep neural network can be represented as a directed acyclic graph with
specific nodes to accept inputs, generate outputs, and perform various
operations to transform inputs into outputs. Circuit analysis involves
localizing and understanding subgraphs within the computational graph of a
model that are responsible for specific behaviors, and has had success in both
language and vision models (Olah et al., 2020; Wang et al., 2022; Räukur et
al., 2022; Chan et al., 2022).

#### Activation Patching.

As introduced in (Meng et al., 2022b), activation patching is a technique that
uses causal intervention to identify which submodules’ activations matter for
producing some model output. The process of activation patching involves
running all the layers of a model until reaching a certain submodule with an
original input, denoted as $A$, and a corrupted input, denoted as $B$. The
activations of this specific submodule with input $B$ are then patched into
the corresponding activations of the same submodule with input $A$ during the
forward pass. Next, the patched activations are fed forward through the rest
of the model. This enables one to assess the role of the specific submodule in
generating an output, by quantifying how much this intervention shifts the
model’s output from its original answer on $A$. Our approach generalizes
activation patching and swaps activation from multiple alternative input
sequence runs with known target outputs, instead of corrupted sequence runs,
as described in (Meng et al., 2022b).

#### Variable Binding.

Variable binding is the process of associating a variable with a specific
value, and is a fundamental concept in symbolic reasoning considered essential
for solving tasks such as natural language understanding and reasoning
(Marcus, 2001). However, it is still a mystery if and how Large Language
Models (LLMs) implement this process.

Please see Appendix B for additional related work.

##  3 Using Desiderata to Localize Computation

We discover circuitry responsible for a specific task by enumerating
properties of such a desired circuitry, and then learning a binary mask over
the model’s parameters which accords with these properties (Fig. 1). We
specify properties (or desiderata) in terms of causal interventions with a
known target effects, and combine various interventions into a single
objective function. We then learn a sparse mask on the targeted model
components, such that applying causal interventions on the masked components
alters model behavior to satisfy the objective function.

#### Model components.

As a first step, we specify our set of model components. Models can be
represented at various levels of granularity. More granular components is more
computational expensive, but allows for more specific localization of a model
behavior. In Section 4, we decompose LLaMA-13B into a set of attention heads
and MLPs, as opposed to more granular (e.g. splitting by Query, Key, and Value
matrices) or less granular (e.g. grouping into layers) representations.

#### Desiderata.

Given a computational circuitry with specific functionality, we define a set
of desiderata to enumerate the effects of various causal interventions on the
circuitry. Each desideratum $d$ corresponds to a set of $n$ 3-tuple, each of
which consists of an original sequence ($o$), an alternate sequence ($a$), and
a target value ($t$). When the activation of the sought-after circuitry
generated with $o$ is replaced with the corresponding activation generated
with $a$, the model should output $t$. The target value ($t$) is determined
based on the nature of the intervention: it can remain equal to the output of
$o$ (indicating no change in the output is expected), be altered to match the
output of $a$, or be set to a completely different third value. We identify
and localize submodule with the desired functionality based on its adherence
to the expected outcomes specified by the desiderata.

Each 3-tuple $(o,a,t)$ contributes to a loss term which measures how well
performing activation patching on a set of model components $\\{c_{i}\\}$
achieves $t$,

$\mathcal{L}_{d}(\\{c_{i}\\})=\frac{1}{n}\sum_{(o,a,t)\in
d}\mathcal{L}(\\{c_{i}\\},(o,a,t),y).$ (1)

Note that some measure of proximity $\mathcal{L}$ between the induced model
output $y$ and the target $t$ is needed. Furthermore, one can combine multiple
desideratum into a single objective function,
$\mathcal{L}_{D}(\\{c_{i}\\})=\sum_{d\in D}\mathcal{L}_{d}(\\{c_{i}\\})$.
Desiderata for the specific case of identifying the value-copying circuitry
involved in variable binding are presented in Fig. 2 and discussed in Section
4.1.

Figure 2: Variable Binding Desiderata. Each desideratum is a set of original
($o$), alternate ($a$), and target ($t$) 3-tuples. In the Value Dependence
desideratum, patching should change the output to the alternate’s output; in
the Operation Invariance desideratum, patching should have no effect.

#### Learning a Binary Mask.

In order to find the set of model components $\\{c_{i}\\}$ that minimizes
$\mathcal{L}_{D}(\\{c_{i}\\})$, we use a continuous relaxation of Equation 1.
We define a mask over the model components by assigning a learnable weight
$w_{i}\in[0,1]$, to each component $c_{i}$. That is, $w_{i}=0$ corresponds to
fully patching component $c_{i}$ with its value $v_{a}$ from the sequence $a$,
$w_{i}=1$ corresponds to not patching $c_{i}$, and $0<w_{i}<1$ corresponds to
taking a convex combination of the $c_{i}$’s activation value $v$ and the
value $v_{a}$

$w_{i}\cdot v+(1-w_{i})\cdot v_{a}.$ (2)

Note that when we patch multiple components of the model, the value $v$ of a
later-layer component will be influenced by the patching of earlier layers
before it itself is combined with $v_{a}$ as above.

We optimize the continuous mask according to $\mathcal{L}_{D}$, which measures
how well the patching intervention defined by the mask meets our desiderata.
We use $\ell_{0.5}$ regularization with tunable strength $\lambda$ over the
mask entries to encourage patching only a sparse set of model components
(Louizos et al., 2018). Throughout learning, we clamp values between 0 and 1.
After training, we round weights to either 0 or 1 to form a binary mask.
Empirically, we find that rounding the mask to become binary typically has
little effect on its ability to satisfy the desiderata, and attribute this to
the regularization for sparsity during training.

##  4 Variable Binding

| VD Acc. (+, -) | OI Acc. (+, -) | VD Acc. ($\times$) | OI Acc. (+, $\times$) | # Patched
---|---|---|---|---|---
Original Model | 18% | 91% | 11% | 93% | 0
Incomplete Desiderata (VD) | 93% | 11% | 82% | 13% | 10
Full Desiderata (VD & OI) | 84% | 82% | 84% | 91% | 10

Table 1: Accuracy of patching experiments. Learning the patching mask
according to an incomplete set of desiderata (only using the Value Dependence
desideratum as presented in the second row) fails to localize our target
computation (Operation Invariance accuracy suffers for all the tested
operation indicated in parenthesis). Using both desiderata (Full Desiderata.
third row) successfully causes Value Dependence behavior while maintaining
Operation Invariance. Interestingly, the learnt patching mask achieves high
accuracy also for operation that was not included in the training set. For all
cases, accuracy is calculated on held-out test set.

We apply our method (Section 3) to locate circuitry responsible for retrieving
variable values when computing simple arithmetic expressions like those in
Fig. 2. We use LLaMA-13B, a 40-layer, decoder-only transformer language model,
trained on a diverse data set (Touvron et al., 2023). We hypothesize that
there exist components of LLaMA-13B that, in order to complete sequences like
those appearing in Fig. 2, copy the value previously assigned to the variable
$x$ into the final token’s residual stream. We further hypothesize that the
$x$’s value is then combined with $y$’s value to compute the desired
expression.

We design desiderata to search specifically for this value-copying circuitry.
Throughout, we only evaluate accuracy of models based on whether their
prediction of the first digit of the answer value is correct; we ensure a
diverse set of targets to avoid degenerate solutions. Code to replicate our
results is available in a public
repository.111https://anonymous.4open.science/r/anima_submission-D770/README.md

###  4.1 Variable Binding Desiderata

We propose two desiderata to isolate this hypothesized value-copying
circuitry:222We note that additional desideratum are possible.

  1. 1.

Value Dependence (VD; Fig. 2, top). Patching our target circuitry with its
activations from alternate sequences containing different $x$ values should
control which value is copied into the final residual stream. Accordingly,
such patching should change the model’s output to match the output of the
alternate sequences.

  2. 2.

Operation Invariance (OI; Fig. 2, bottom). Since we are looking for circuitry
shared across arithmetic operations, the specific operation being performed in
the expression should not affect the behavior of the value-copying circuitry,
as it should copy the same variable value regardless of the operation. We
therefore form alternate sequences with a flipped operation (either addition
or subtraction), with a target value equals to the output value of the
corresponding original sequence.

###  4.2 Binary Mask Details

We consider all MLPs and attention heads (1640 models’ components in total)
and learn a binary mask as described in Section 3. We only perform patching to
each component’s contribution to the final-token residual stream, as that is
where we expect the value-copying circuitry to be active. We use two-digit
variable values with addition and subtraction operations for defining both VD
and OI sequences. We use the logit difference between the original and
alternate answers as the proximity measure. For the VD task, we intend to
maximize the logit difference, whereas, for the OI task, we aim to minimize
it.

We created a dataset comprising VD and OI sequences, with a total of $90/90$
train/test examples, such that the first digit of the expected answer is
uniformly drawn from $[1,9]$. We use the Adam optimizer (Kingma & Ba, 2017)
with a learning rate of 0.01, and alternate between taking gradient steps from
the VD loss and the OI loss to save memory. For all experiments described
below we use a sparsity regularization weight of $\lambda=0.03$. In Appendix A
we present additional results with different experimental settings such as
varying $\lambda$ and the numbers of patched attention heads.

##  5 Results

According to our desiderata, we have identified a set of ten components,
comprising nine attention heads and one MLP, that execute variable binding. We
patch heads according to this ten-component mask and evaluate the model’s
accuracy on a held-out set of VD and OI problems. On VD scenarios, accuracy
measures how often the model outputs the answer from the alternative sequence.
On OI scenarios, accuracy measures how often the model outputs the answer from
the original sequence. We expect a mask that finds heads corresponding to the
value copying subtask of variable binding to achieve high accuracy in both VD
and OI.

We observe that the models’ components identified by our method exhibit high
accuracy in both tasks, as indicated in Table 1 (first and second columns). We
further test these components on VD problems involving a multiplication
operation instead of addition or subtraction, and on OI problems involving
swapping between addition and multiplication. Surprisingly, we find that the
accuracy remains high in these scenarios, despite not including multiplication
during training (see Table 1, third and fourth columns). This indicates that
these ten components indeed serve as the circuitry that copies variable values
to the final residual stream (before the model operates on them); these
components successfully cause the model’s output to change in the case that
one of the bound values changes, but not in the case that the operation in the
equation changes, even when testing an operation that was not include in
training.

We also find that including both desiderata is crucial for locating this
circuitry; with only the VD desideratum, the identified heads successfully
alter model behavior in the VD scenario, but also affect the model’s output in
the OI scenario (see Table 1, second row). When both desiderata are included
in the loss, the masked components are mostly attention heads in the middle of
the model;333Heads 11.11, 12.0, 12.7, 15.11, 15.25, 17.17, 18.11, 18.18,
19.20, and MLP 27. whereas when only using the first desideratum, the masked
components form a cluster of late-layer MLPs444MLPs 18, 27, 28, 29, 30, 31,
32, 33, 35, and 36.. A possible explanation for this is that using only the VD
desideratum, the mask includes model components that write the computed final
value of the expression to the residual stream, whereas adding the second
desideratum encourages the mask to find the value-copying circuitry.

##  6 Conclusion

In this paper, we proposed a new approach to localizing model components
responsible for performing a specific task, using a set of causal behavior
desiderata. Our method localize 10 components responsible for copying variable
values in LLaMA-13B. We plan to compare it with existing localization methods
(Meng et al., 2022a; Conmy et al., 2023) and to expend it to additional tasks.

## Acknowledgments

XD and MN would like to thank Sam Marks and Oam Patel for their valuable
discussions. NP and DB are supported by grants from Open Philanthropy. TRS is
supported in part by the Zuckerman STEM Leadership Program and the Viterbi
Fellowship.

## References

  * Burns et al. (2022) Burns, C., Ye, H., Klein, D., and Steinhardt, J.  Discovering latent knowledge in language models without supervision, 2022\.
  * Carroll et al. (2023) Carroll, M., Chan, A., Ashton, H., and Krueger, D.  Characterizing manipulation from ai systems, 2023.
  * Chan et al. (2022) Chan, L., Garriga-Alonso, A., Goldowsky-Dill, N., Greenblatt, R., Nitishinskaya, J., Radhakrishnan, A., Shlegeris, B., and Thomas, N.  Causal Scrubbing: a method for rigorously testing interpretability hypotheses [Redwood Research], December 2022.  URL https://www.alignmentforum.org/posts/JvZhhzycHu2Yd57RN/causal-scrubbing-a-method-for-rigorously-testing.
  * Conmy et al. (2023) Conmy, A., Mavor-Parker, A. N., Lynch, A., Heimersheim, S., and Garriga-Alonso, A.  Towards automated circuit discovery for mechanistic interpretability, 2023\.
  * Geiger et al. (2022) Geiger, A., Wu, Z., Lu, H., Rozner, J., Kreiss, E., Icard, T., Goodman, N., and Potts, C.  Inducing causal structure for interpretable neural networks.  In _International Conference on Machine Learning_ , pp. 7324–7338. PMLR, 2022.
  * Geiger et al. (2023) Geiger, A., Wu, Z., Potts, C., Icard, T., and Goodman, N. D.  Finding alignments between interpretable causal variables and distributed neural representations.  _arXiv preprint arXiv:2303.02536_ , 2023.
  * Kingma & Ba (2017) Kingma, D. P. and Ba, J.  Adam: A method for stochastic optimization, 2017.
  * Li et al. (2022) Li, K., Hopkins, A. K., Bau, D., Viégas, F., Pfister, H., and Wattenberg, M.  Emergent world representations: Exploring a sequence model trained on a synthetic task, 2022.  URL https://arxiv.org/abs/2210.13382.
  * Louizos et al. (2018) Louizos, C., Welling, M., and Kingma, D. P.  Learning Sparse Neural Networks through $L_0$ Regularization, June 2018.  URL http://arxiv.org/abs/1712.01312.  arXiv:1712.01312 [cs, stat].
  * Marcus (2001) Marcus, G. F.  Relations between Variables.  In _The Algebraic Mind: Integrating Connectionism and Cognitive Science_. The MIT Press, 04 2001.  ISBN 9780262279086.  doi: 10.7551/mitpress/1187.003.0005.  URL https://doi.org/10.7551/mitpress/1187.003.0005.
  * Meng et al. (2022a) Meng, K., Bau, D., Andonian, A., and Belinkov, Y.  Locating and editing factual associations in gpt.  _Advances in Neural Information Processing Systems_ , 35:17359–17372, 2022a.
  * Meng et al. (2022b) Meng, K., Bau, D., Andonian, A., and Belinkov, Y.  Locating and editing factual associations in GPT.  _Advances in Neural Information Processing Systems_ , 36, 2022b.
  * Olah et al. (2020) Olah, C., Cammarata, N., Schubert, L., Goh, G., Petrov, M., and Carter, S.  Zoom in: An introduction to circuits.  _Distill_ , 2020.  doi: 10.23915/distill.00024.001.  https://distill.pub/2020/circuits/zoom-in.
  * Perez et al. (2022) Perez, E., Ringer, S., Lukošiūtė, K., Nguyen, K., Chen, E., Heiner, S., Pettit, C., Olsson, C., Kundu, S., Kadavath, S., Jones, A., Chen, A., Mann, B., Israel, B., Seethor, B., McKinnon, C., Olah, C., Yan, D., Amodei, D., Amodei, D., Drain, D., Li, D., Tran-Johnson, E., Khundadze, G., Kernion, J., Landis, J., Kerr, J., Mueller, J., Hyun, J., Landau, J., Ndousse, K., Goldberg, L., Lovitt, L., Lucas, M., Sellitto, M., Zhang, M., Kingsland, N., Elhage, N., Joseph, N., Mercado, N., DasSarma, N., Rausch, O., Larson, R., McCandlish, S., Johnston, S., Kravec, S., Showk, S. E., Lanham, T., Telleen-Lawton, T., Brown, T., Henighan, T., Hume, T., Bai, Y., Hatfield-Dodds, Z., Clark, J., Bowman, S. R., Askell, A., Grosse, R., Hernandez, D., Ganguli, D., Hubinger, E., Schiefer, N., and Kaplan, J.  Discovering language model behaviors with model-written evaluations, 2022\.
  * Räukur et al. (2022) Räukur, T., Ho, A., Casper, S., and Hadfield-Menell, D.  Toward transparent ai: A survey on interpreting the inner structures of deep neural networks.  _arXiv preprint arXiv:2207.13243_ , 2022.
  * Touvron et al. (2023) Touvron, H., Lavril, T., Izacard, G., Martinet, X., Lachaux, M.-A., Lacroix, T., Rozière, B., Goyal, N., Hambro, E., Azhar, F., Rodriguez, A., Joulin, A., Grave, E., and Lample, G.  Llama: Open and efficient foundation language models, 2023.  URL https://arxiv.org/abs/2302.13971.
  * Vig et al. (2020) Vig, J., Gehrmann, S., Belinkov, Y., Qian, S., Nevo, D., Singer, Y., and Shieber, S.  Investigating gender bias in language models using causal mediation analysis.  _Advances in neural information processing systems_ , 33:12388–12401, 2020.
  * Wang et al. (2022) Wang, K., Variengien, A., Conmy, A., Shlegeris, B., and Steinhardt, J.  Interpretability in the wild: a circuit for indirect object identification in gpt-2 small, 2022.
  * Wu et al. (2023) Wu, Z., Geiger, A., Potts, C., and Goodman, N. D.  Interpretability at scale: Identifying causal mechanisms in alpaca, 2023\.

##  Appendix A Varying regularization strength

We vary the regularization strength $\lambda$ in order to learn masks with
varying numbers of heads. We find that setting the regularization so that the
masks learns approximately 10 model subcomponents is the approximate minimum
number that can score highly for held-out Value Dependence accuracy and
Operation Invariance accuracy, and so we use that setting for the main results
of our paper. We also show further that removing the Operation Invariance
desideratum causes the mask to score poorly on that criterion.

Figure 3: Evaluating masks of various numbers of heads on held-out VD and OI
problems. Each vertical pair of datapoints corresponds to a mask learned by a
training run with a different value of $\lambda$, the sparsity regularization
weight. With too few components patched, the model does not score well at
Value Dependence. We interpret this as indicating that not enough of the
value-copying heads have been patched. Figure 4: Transfer to accuracy on
multiplication problems. This graph depicts the same masks as Fig. 3 (which
were trained on sequences involving only addition and subtraction), but
evaluated on all-multiplication Value Dependence problems, and addition-to-
multiplication (and vice versa) Operation Invariance problems. Similarly to
Fig. 3, VD accuracy is low with too few heads patched. Figure 5: Varying
regularization strength with incomplete desiderata. This graph demonstrates
learning a mask with only the Value Dependence desideratum. Again, each
vertical pair of datapoints corresponds to a mask learned by a training run
with a different value of $\lambda$, the sparsity regularization weight.
Unlike when the mask is optimized according to both desiderata, these masks
fail to achieve high accuracies on both Operation Invariance and Value
dependence at the same time, as discussed in Section 4.

##  Appendix B Related Work

Previous work has developed automated approaches to localizing computation
(Conmy et al., 2023; Geiger et al., 2022; Wu et al., 2023). Our work varies
from Conmy et al. (2023) in learning a mask and considering a broader class of
ablations (patches to change behavior, instead of just preserve). Our work
shares features with recent work from Geiger et al. (2023) and Wu et al.
(2023), but differs in attempting to isolate shared computation common in
multiple input-output circuits as opposed to understanding full input-output
circuits.