††thanks: Corresponding author

# Investigation of $P_{cs}(4459)^{0}$ pentaquark via its strong decay to
$\Lambda J/\psi$

K. Azizi<EMAIL_ADDRESS>Department of Physics, University of Tehran,
North Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doğuş
University, Acıbadem-Kadıköy, 34722 Istanbul, Turkey  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  Y. Sarac<EMAIL_ADDRESS>Electrical and
Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey
H. Sundu<EMAIL_ADDRESS>Department of Physics, Kocaeli
University, 41380 Izmit, Turkey

###### Abstract

Recently the observation of a new pentaquark state, the hidden-charmed strange
$P_{cs}(4459)^{0}$, was reported by the LHCb Collaboration. The spin-parity
quantum numbers of this state were not determined as a result of insufficient
statistics. To shed light on its quantum numbers, we investigate its decay,
$P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$, the mode that this state has been
observed, within the QCD sum rule framework. We obtain the width of this decay
assigning the spin-parity quantum numbers of $P_{cs}(4459)^{0}$ state as
$J^{P}=\frac{1}{2}^{-}$ and its substructure as diquark-diquark-antiquark. To
this end, we first calculate the strong coupling constants defining the
considered decay and then use them in the width calculations. The obtained
width is consistent with the experimental observation, confirming the quantum
numbers $J^{P}=\frac{1}{2}^{-}$ and compact pentaquark nature for
$P_{cs}(4459)^{0}$ state.

##  I Introduction

In the past two decades, starting with the observation of the $X(3872)$
Choi:2003ue , we witnessed the observations of many exotic hadrons candidates
for tetraquarks Zyla:2020zbs  and pentaquarks Aaij:2015tga ; Aaij:2016ymb ;
Aaij:2019vzc . The first observation of pentaquark states was announced in
2015 by the LHCb collaboration Aaij:2015tga  and two pentaquark states in
$J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\rightarrow J/\psi
pK^{-}$ decays were reported with the following resonance parameters
Aaij:2015tga : $m_{P_{c}(4380)^{+}}=4380\pm 8\pm 29~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4380)^{+}}=205\pm 18\pm 86~{}\mathrm{MeV}$ and
$m_{P_{c}(4450)^{+}}=4449.8\pm 1.7\pm 2.5~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4450)^{+}}=39\pm 5\pm 19~{}\mathrm{MeV}$. The LHCb
Collaboration supported this observation later, in 2016, with a full amplitude
analysis for $\Lambda_{b}^{0}\rightarrow J/\psi p\pi^{-}$ decays Aaij:2016ymb
. In 2019, a new pentaquark resonance, $P_{c}(4312)^{+}$, was reported by the
LHCb Collaboration with the following mass and width Aaij:2019vzc :
$m_{P_{c}(4312)^{+}}=4311.9\pm 0.7^{+6.8}_{-0.6}~{}\mathrm{MeV}$ and
$\Gamma_{P_{c}(4312)^{+}}=9.8\pm 2.7^{+3.7}_{-4.5}~{}\mathrm{MeV}$. Together
with the $P_{c}(4312)^{+}$ state, the LHCb also announced the split of the
peak corresponding to $P_{c}(4450)^{-}$ into two peaks which have the
following masses and widths: $m_{P_{c}(4440)^{+}}=4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4440)^{+}}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mathrm{MeV}$ and $m_{P_{c}(4457)^{+}}=4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4457)^{+}}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mathrm{MeV}$ Aaij:2019vzc . These observations and the
advances in experimental facilities and techniques indicate the possibility to
observe more exotic states in the future.

On the other hand, there is still uncertainties in the sub-structures and
quantum numbers of these observed pentaquark states. In that matter, there are
different proposals and theoretical works about these resonances in the
literature analyzing their parameters and giving consistent predictions with
their observed properties. It is obvious that deeper investigations are
required not only to differentiate these proposals but also to help better
identify the nature of these states. Understanding the inner structures and
properties of these exotic states may also support their future
investigations. Besides, they may provide improvements in understanding the
dynamics of the quantum chromodynamics (QCD) in its nonperturbative domain.
With their non-conventional quark substructures that are different from the
conventional baryons composed of three quarks/antiquarks or mesons composed of
a quark and an antiquark, they provide an attractive ground for the
understanding of the nonperturbative nature of strong interaction. Although
the investigations of such exotic states extend before their observations,
with their observations the pentaquarks have become a hot topic in all these
respects. With these motivations and the excitement brought by their
observations, their various properties were investigated widely with different
approaches to shed light on their nonspecific sub-structures and quantum
numbers. Based on their close masses to meson-baryon threshold, they were
assigned as meson baryon molecular states in Refs. Chen:2015loa ; Chen:2015moa
; He:2015cea ; Meissner:2015mza ; Roca:2015dva ; Azizi:2016dhy ; Azizi:2018bdv
; Azizi:2020ogm ; Chen:2020opr . They were interpreted with diquark-diquark-
antiquark Lebed:2015tna ; Li:2015gta ; Maiani:2015vwa ; Anisovich:2015cia ;
Wang:2015ava ; Wang:2015epa ; Wang:2015ixb ; Ghosh:2015ksa ; Wang:2015wsa ;
Zhang:2017mmw ; Wang:2019got ; Wang:2020rdh ; Ali:2020vee ; Wang:2016dzu  and
diquark-triquark Wang:2016dzu ; Zhu:2015bba  models. To investigate their
properties in Ref. Liu:2017xzo  a variant of the D4-D8 model and in Ref.
Scoccola:2015nia  the topological soliton model were used. They were also
explained as kinematical effects Guo1 ; Guo2 ; Mikhasenko:2015vca ; Liu1 ;
Bayar:2016ftu . Besides the observed ones, the possible other candidate
pentaquark states were also considered in the literature with different quark
contents Liu:2020cmw ; Chen:2015sxa ; Feijoo:2015kts ; Lu:2016roh ;
Irie:2017qai ; Chen:2016ryt ; Zhang:2020cdi ; Paryev:2020jkp ; Gutsche:2019mkg
; Azizi:2017bgs ; Cao:2019gqo ; Azizi:2018dva ; Zhang:2020vpz ; Wang:2020bjt ;
Xie:2020ckr .

Recently, in a talk, implications of LHCb measurements and future prospects,
the evidence for a pentaquark including a strange quark in its quark content
was first announced by the LHCb Collaboration Wang111  and later it was
reported in the Ref. Aaij:2020gdg . The $P_{cs}(4459)^{0}$ was observed in
$\Xi_{b}^{-}\rightarrow J/\psi K^{-}\Lambda$ decays with the following
measured mass and width Aaij:2020gdg :

$\displaystyle M=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mathrm{MeV},$ (1)

with statistical significance exceeding $3\sigma$ and there is no
determination for its spin parity quantum numbers, yet. With a mass just below
$\Sigma_{c}\bar{D}^{*}$ threshold, the $P_{cs}(4459)^{0}$ was interpreted as
$\bar{D}^{*}\Sigma_{c}$ hadronic molecular state in Ref. Chen:2020uif . The
analyses were conducted using QCD sum rule method and the results supported
its possibility to be $\bar{D}^{*}\Sigma_{c}$ molecular state with either
$J^{P}=\frac{1}{2}^{-}$ or $J^{P}=\frac{3}{2}^{-}$ giving mass values
consistent with the experimentally reported one Chen:2020uif . Molecular
explanation for the $P_{cs}(4459)^{0}$ was also discussed in Ref. Peng:2020hql
using effective field formalism and the masses were predicted considering the
$\bar{D}^{*}\Xi_{c}$ molecular pictures with $J^{P}=\frac{1}{2}$ and
$J=\frac{3}{2}$ as $4469$ MeV and $4453-4463$ MeV, respectively. With these
results the spin of the $P_{cs}(4459)^{0}$ state was suggested to possibly be
$J=\frac{3}{2}$. In Ref. Chen:2020kco  molecular interpretation was taken into
account using one-boson-exchange model and $P_{cs}(4459)^{0}$ was interpreted
as a coupled
$\Xi_{c}\bar{D}^{*}/\Xi_{c}^{*}\bar{D}/\Xi_{c}^{{}^{\prime}}\bar{D}^{*}/\Xi_{c}^{*}\bar{D}^{*}$
bound state that has $I(J^{P})=0(\frac{3}{2}^{-})$. In Ref. Wang:2020eep  the
mass analysis was made via QCD sum rule approach for a pentaquark state
containing strange quark with an interpolating current in the scalar-diquark-
scalar-diquark-antiquark form. Based on the mass value obtained for the state
as $M=4.47\pm 0.11$ MeV, which was consistent with the experimentally observed
one, $P_{cs}(4459)^{0}$ was assigned to have the quantum numbers
$J^{P}=\frac{1}{2}^{-}$.

As is seen, the quantum numbers for the $P_{cs}(4459)^{0}$ state were not
determined by the experiment, and from different studies there are different
assumptions for its quantum numbers and sub-structure, indicating the
necessity for further investigations of the properties of this state. Inspired
by this, we investigate the $P_{cs}(4459)^{0}$ state through its strong decay
via QCD sum rule method Shifman:1978bx ; Shifman:1978by ; Ioffe81 . This
method has a wide range of applications in the literature, which resulted in
successful predictions consistent with the experimental observations. To
analyze the pentaquark states within the QCD sum rule approach a proper choice
of the interpolating current is necessary. So far, it has been observed that,
in various QCD sum rules analyses for observed pentaquark states, the
different choices of the interpolating fields, either in the molecular form or
in the diquark-diquark antiquark form, were applied. These analyses have
resulted in mass predictions that are consistent with the experimental
observations. In Refs. Wang:2021itn ; Wang:2020rdh , it was pointed out that
the hadronic dressing mechanism, which also works for $X$, $Y$, $Z$ states
Wang:2019iaa , may compromise the interpretation of these states as a molecule
or diquark-diquark-antiquark states considering the result of the QCD sum
rules. This may be attributed to the possibility that the pentaquark states
may have both the diquark-diquark-antiquark and meson-baryon type Fock
components. The pentaquark states may have a typical size of the conventional
baryon with a diquark-diquark-antiquark type kernel, and the strong couplings
to the meson-baryon pairs may cause spending a considerable part of their life
time as meson-baryon molecules. The local interpolating current in the
diquark-diquark-antiquark form can be formed from special superposition of the
meson-baryon type interpolating currents and the opposite is also possible,
that is a meson baryon current can also be written as a special superposition
of diquark-diquark-antiquark type currents that carries the net effect, (see
for instance Ref. Wang:2019got ). To interpolate the pentaquark state, one can
choose either diquark-diquark-antiquark or molecular type currents. Any
current type with the same quark structure and quantum numbers of the
pentaquark’s Fock states may couple to the pentaquark. The main component of
the Fock states may give the sub-structure of the pentaquark. For more
details, we refer to the Refs. Wang:2021itn ; Wang:2020rdh ; Wang:2018waa ;
Wang:2019iaa ; Wang:2019hyc . Therefore, besides mass predictions, the
investigations of their decay mechanisms, using different choices for
interpolating currents, may help to distinguish their substructure with
comparisons of the results to experimental results. In this work, to provide
the width, we first calculate the strong coupling constants defining the decay
$P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$ using three-point QCD sum rule
approach with an interpolating current in the scalar-diquark-scalar-diquark-
antiquark form of $J^{P}=\frac{1}{2}^{-}$. Then, the obtained results for the
strong coupling constants are used to determine the corresponding width value.
We compare the obtained result with the experimental observation to shed light
on the quantum numbers and quark sub-structure of the considered state.

The organization of the paper is as follows: In next section we give the
details of the QCD sum rule calculations for the strong coupling constants
defining the $P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$ decay. The numerical
analyses of the obtained sum rules as well as the width of the considered
decay are also presented in Sec. II. Last section is devoted to a summary and
comparison of the obtained result for the width to that of the experiment.

##  II The strong decay $P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$

In this section the details of the calculations for the strong coupling
constants and the width of the strong decay $P_{cs}(4459)^{0}\rightarrow
J/\psi\Lambda$ and their numerical analyses are given. The correlation
function required for the calculations has the following form:

$\Pi_{\mu}(p,q)=i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\langle
0|\mathcal{T}\\{\eta^{\Lambda}(y)\eta_{\mu}^{J/\psi}(0)\bar{\eta}^{P_{cs}}(x)\\}|0\rangle,$
(2)

where the $\eta^{P_{cs}}$, $\eta^{\Lambda}$ and $\eta^{J/\psi}$ are the
interpolating currents of the considered states which have the same quantum
numbers with these states and $\mathcal{T}$ is used to represent the time
ordering operator. The interpolating currents are given as:

$\displaystyle\eta^{P_{cs}}$ $\displaystyle=$
$\displaystyle\epsilon^{ila}\epsilon^{ijk}\epsilon^{lmn}u^{T}_{j}C\gamma_{5}d_{k}s^{T}_{m}C\gamma_{5}c_{n}C\bar{c}^{T}_{a},$
$\displaystyle\eta^{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\epsilon^{lmn}\sum_{i=1}^{2}\Big{[}2(u^{T}_{l}CA_{1}^{i}d_{m})A_{2}^{i}s_{n}+(u^{T}_{l}CA_{1}^{i}s_{m})A_{2}^{i}d_{n}+(d^{T}_{n}CA_{1}^{i}s_{m})A_{2}^{i}u_{l}\Big{]},$
$\displaystyle\eta_{\mu}^{J/\psi}$ $\displaystyle=$
$\displaystyle\bar{c}_{l}i\gamma_{\mu}c_{l},$ (3)

where the sub-indices, $a,~{}i,~{}j,~{}k,~{}l,~{}m,~{}n$ represent the color
indices $u,~{}d,~{}s,~{}c$ are the quark fields, $C$ is charge conjugation
operator; and $A_{1}^{1}=I$, $A_{1}^{2}=A_{2}^{1}=\gamma_{5}$ and
$A_{2}^{2}=\beta$ is an arbitrary parameter to be determined from the
analyses. The above correlation function is calculated in two representations
which are called hadronic and QCD representations. The QCD sum rules for the
physical quantities are obtained from the matches of the coefficients of the
same Lorentz structures attained on both sides.

In the hadronic representation of the correlation function, the interpolating
currents are treated as operators creating or annihilating the hadronic
states. To proceed in the calculation of this side, complete sets of related
hadronic states that have the same quantum numbers with the given
interpolating currents are inserted inside the correlator. After taking four
integrals the results turn into

$\displaystyle\Pi_{\mu}^{\mathrm{Had}}(p,q)=\frac{\langle
0|\eta^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle\langle
0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle\langle
P_{cs}(p,s)|\eta^{P_{cs}}|0\rangle}{(m_{\Lambda}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-q^{2})(m_{P_{cs}}^{2}-p^{2})}+\cdots,$ (4)

where $\cdots$ is used to represent the contributions of higher states and
continuum, the $p$, $p^{\prime}$ and $q$ are the momenta of the $P_{cs}$ and
$\Lambda$ and $J/\psi$ states, respectively. The matrix elements in this
result are defined in terms of the masses and current coupling constants, and
they have the following forms:

$\displaystyle\langle 0|\eta^{P_{cs}}|P_{cs}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{P_{cs}}u_{P_{cs}}(p,s).$ $\displaystyle\langle
0|\eta^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{\Lambda}u_{\Lambda}(p^{\prime},s^{\prime}),$
$\displaystyle\langle 0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle$ $\displaystyle=$
$\displaystyle f_{J/\psi}m_{J/\psi}\varepsilon_{\mu},$ (5)

where $\varepsilon_{\mu}$ is the polarization vector and $f_{J/\psi}$ is the
decay constant of the $J/\psi$ state, $\lambda_{P_{cs}}$, $\lambda_{\Lambda}$
are the current coupling constants of the $P_{cs}$ and $\Lambda$ states,
$u_{P_{cs}}$ and $u_{\Lambda}$ are the corresponding spinors, respectively.
$|P_{cs}(p,s)\rangle$ is used to represent one-particle pentaquark state with
negative parity. The matrix element $\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle$ is given in terms
of the coupling constants, $g_{1}$ and $g_{2}$ as

$\displaystyle\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle=\epsilon^{*\nu}\bar{u}_{\Lambda}(p^{\prime},s^{\prime})\big{[}g_{1}\gamma_{\nu}-\frac{i\sigma_{\nu\alpha}}{m_{\Lambda}+m_{P_{cs}}}q^{\alpha}g_{2}\big{]}\gamma_{5}u_{P_{cs}}(p,s).$
(6)

In the next step, the matrix elements given in Eqs. (5) and (6) are placed in
the Eq. (4) and following summations over spins of spinors and polarization
vectors are applied

$\displaystyle\sum_{s}u_{P_{cs}}(p,s)\bar{u}_{P_{cs}}(p,s)$ $\displaystyle=$
$\displaystyle({\not{p}}+m_{P_{cs}}),$
$\displaystyle\sum_{s^{\prime}}u_{\Lambda}(p^{\prime},s^{\prime})\bar{u}_{\Lambda}(p^{\prime},s^{\prime})$
$\displaystyle=$ $\displaystyle({\not{p}^{\prime}}+m_{\Lambda}),$
$\displaystyle\varepsilon_{\alpha}\varepsilon^{*}_{\beta}$ $\displaystyle=$
$\displaystyle-g_{\alpha\beta}+\frac{q_{\alpha}q_{\beta}}{m_{J/\psi}^{2}}.$
(7)

And finally, after the Borel transformation, which is applied to suppress the
contributions coming from higher states and continuum, the result of physical
side is obtained as

$\displaystyle\tilde{\Pi}_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle
e^{-\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{-\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}}{m_{J/\psi}(m_{\Lambda}+m_{P_{cs}})(m_{J/\psi}^{2}+Q^{2})}\big{[}-g_{1}(m_{\Lambda}+m_{P_{cs}})^{2}+g_{2}m_{J/\psi}^{2}\big{]}\not\\!pp_{\mu}\gamma_{5}$
(8) $\displaystyle+$ $\displaystyle
e^{-\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{-\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{J/\psi}m_{\Lambda}}{(m_{\Lambda}+m_{P_{cs}})(m_{J/\psi}^{2}+Q^{2})}\big{[}g_{1}(m_{\Lambda}+m_{P_{cs}})+g_{2}(m_{\Lambda}-m_{P_{cs}})\big{]}\not\\!p\gamma_{\mu}\gamma_{5}$
$\displaystyle+$ $\displaystyle\mathrm{other~{}structures}+\cdots$

where $M^{2}$ and $M^{\prime 2}$ are the Borel parameters to be determined
from the analyses imposing some necessary criteria and $Q^{2}=-q^{2}$. The
result contains more Lorentz structures than the ones given explicitly in Eq.
(8). However, in the last equation, we present only the ones that are used
directly in the analyses, and the others and the contribution of the excited
states and continuum are represented as $\mathrm{other~{}structures}+\cdots$.

The second representation of the correlation function, the QCD side, is
obtained using the interpolating currents explicitly in the correlation
function. To this end, the possible contractions between the quark fields are
attained using Wick’s theorem that renders the result to the one given in
terms of heavy and light quark propagators as:

$\displaystyle\Pi_{\mu}^{\mathrm{QCD}}(p,p^{\prime},q)$ $\displaystyle=$
$\displaystyle i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\epsilon^{klm}\epsilon^{i^{\prime}l^{\prime}a^{\prime}}\epsilon^{i^{\prime}j^{\prime}k^{\prime}}\epsilon^{l^{\prime}m^{\prime}n^{\prime}}\frac{1}{\sqrt{6}}\sum_{i=1}^{2}\bigg{\\{}-2Tr[\gamma_{5}CS_{u}^{Tkj^{\prime}}(y-x)CA_{1}^{i}S_{d}^{lk^{\prime}}(y-x)]$
(9) $\displaystyle\times$ $\displaystyle
A_{2}^{i}S_{s}^{mm^{\prime}}(y-x)\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C+A_{2}^{i}S_{d}^{mk^{\prime}}(y-x)\gamma^{5}CS_{u}^{Tkj^{\prime}}(y-x)CA_{1}^{i}S_{s}^{lm^{\prime}}(y-x)$
$\displaystyle\times$
$\displaystyle\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C+A_{2}^{i}S_{u}^{kj^{\prime}}(y-x)\gamma^{5}CS_{d}^{mk^{\prime}}(y-x)CA_{1}^{i}S_{s}^{lm^{\prime}}(y-x)\gamma_{5}C$
$\displaystyle\times$ $\displaystyle
S_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C\bigg{\\}},$

where $S_{q}^{ab}(x)=S_{u,d,s}^{ab}(x)$ and $S_{c}^{ab}(x)$ are the light and
heavy quark propagators with the following explicit expressions:

$\displaystyle S_{q}^{ab}(x)=$ $\displaystyle
i\frac{x\\!\\!\\!/}{2\pi^{2}x^{4}}\delta_{ab}-\frac{m_{q}}{4\pi^{2}x^{2}}\delta_{ab}-\frac{\langle\overline{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}}{4}x\\!\\!\\!/\Big{)}\delta_{ab}-\frac{x^{2}}{192}m_{0}^{2}\langle\overline{q}q\rangle\Big{(}1-i\frac{m_{q}}{6}x\\!\\!\\!/\Big{)}\delta_{ab}$
$\displaystyle-\frac{ig_{s}G_{ab}^{\theta\eta}}{32\pi^{2}x^{2}}\Big{[}x\\!\\!\\!/\sigma_{\theta\eta}+\sigma_{\theta\eta}x\\!\\!\\!/\Big{]}-\frac{x\\!\\!\\!/x^{2}g_{s}^{2}}{7776}\langle\overline{q}q\rangle^{2}\delta_{ab}+\cdots,$

and

$\displaystyle S_{c}^{ab}(x)=$ $\displaystyle\frac{i}{(2\pi)^{4}}\int
d^{4}ke^{-ik.x}\Big{\\{}\frac{\delta_{ab}}{k\\!\\!\\!/-m_{c}}-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}(k\\!\\!\\!/+m_{c})+(k\\!\\!\\!/+m_{c})\sigma_{\alpha\beta}}{(k^{2}-m_{c}^{2})^{2}}$
(11)
$\displaystyle+\frac{\pi^{2}}{3}\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\delta_{ab}m_{c}\frac{k^{2}+m_{c}k\\!\\!\\!/}{(k^{2}-m_{c}^{2})^{4}}+\cdots\Big{\\}}.$

The same Lorentz structures obtained in the hadronic side are also present in
this side, and the ones used in our analyses are $\not\\!pp_{\mu}\gamma_{5}$
and $\not\\!p\gamma_{\mu}\gamma_{5}$, whose contributions are represented in
the below equation explicitly, and the contributions of the others are
represented with the last term stated as $\mathrm{other\,\,\,structures}$.

$\displaystyle\Pi_{\mu}^{QCD}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}\,\not\\!pp_{\mu}\gamma_{5}+\Pi_{2}\,\not\\!p\gamma_{\mu}\gamma_{5}+\mathrm{other\,\,\,structures}.$
(12)

To obtain the coefficients, $\Pi_{i}$, of these Lorentz structures, we use the
propagators explicitly in Eq. (9) and transform the results into momentum
space. After computation of the four integrals the spectral densities,
$\rho_{i}$ are obtained from the imaginary part of the results,
$\rho_{i}(s,s^{\prime},q^{2})=\frac{1}{\pi}Im[\Pi_{i}]$. These spectral
densities are used in the following dispersion relation:

$\displaystyle\Pi_{i}=\int ds\int
ds^{\prime}\frac{\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})+\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})}{(s-p^{2})(s^{\prime}-p^{\prime 2})},$ (13)

giving us the final results of the QCD representation of the correlation
function. In the last equation $i=1,2,..,12$ and
$\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})$ and $\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})$ represent the perturbative and non-perturbative
parts of the spectral densities, respectively. The results of the spectral
densities that are used in the analyses ($i=1,2$) are:

$\displaystyle\rho_{1}^{\mathrm{pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{1024\sqrt{6}\pi^{6}\chi^{3}\chi^{\prime
6}}(1+5\beta)m_{s}xy(Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2})^{2}\Theta[L(s,s^{\prime},Q^{2},x,y)],$
$\displaystyle\rho_{1}^{\mathrm{non-pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\Bigg{\\{}-\frac{1}{128\sqrt{6}\pi^{4}\chi^{2}\chi^{\prime
5}}\big{[}\big{(}(\beta-1)\langle\bar{d}d\rangle+\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}xy\big{(}Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2}\big{)}\big{]}$ (14) $\displaystyle-$
$\displaystyle\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{36864\sqrt{6}\pi^{4}\chi^{4}\chi^{\prime
5}}(1+5\beta)m_{c}x^{4}y^{2}+\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{(9216\sqrt{6}\pi^{4}\chi^{3}\chi^{\prime
4})}(1+5\beta)m_{s}xy\big{[}9x^{2}+9(y-1)^{2}+x(19y-18)\big{]}$
$\displaystyle+$ $\displaystyle\frac{1}{256\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}\big{[}m_{0}^{2}\big{(}(\beta-1)\langle\bar{d}d\rangle+\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}xy\big{]}\Bigg{\\}}\Theta[L(s,s^{\prime},q^{2},x,y)],$

and

$\displaystyle\rho_{2}^{\mathrm{pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{2048\sqrt{6}\pi^{6}\chi^{3}\chi^{\prime
4}}\Big{[}-(1+5\beta)m_{c}m_{s}\big{(}Q^{2}xy+m_{c}^{2}\chi^{\prime
2}+s^{\prime}(x^{2}+(y-1)y+x(2y-1))\big{)}^{2}\Big{]}$ $\displaystyle\times$
$\displaystyle\Theta[L(s,s^{\prime},Q^{2},x,y)]$
$\displaystyle\rho_{2}^{\mathrm{non-pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\Bigg{\\{}\frac{1}{512\sqrt{6}\pi^{4}\chi^{2}\chi^{\prime
4}}\Big{[}m_{s}\big{(}\langle\bar{s}s\rangle(1+5\beta)-2(\beta-1)\langle\bar{u}u\rangle\big{)}\chi\big{(}2Q^{2}xy+s^{\prime}\chi\chi^{\prime}\big{)}-2m_{c}\big{(}\langle\bar{s}s\rangle(1+5\beta)$
(15) $\displaystyle+$
$\displaystyle(\beta-1)\langle\bar{u}u\rangle\big{)}\chi^{\prime}\big{(}Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2}\big{)}-2(\beta-1)\langle\bar{d}d\rangle\big{(}m_{s}\chi(2Q^{2}xy+s^{\prime}\chi\chi^{\prime})+m_{c}\chi^{\prime}(Q^{2}xy+s^{\prime}\chi\chi^{\prime}$
$\displaystyle+$ $\displaystyle m_{c}^{2}\chi^{\prime
2})\big{)}\big{]}+\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{147456\sqrt{6}\pi^{4}\chi^{4}\chi^{\prime
4}}(1+5\beta)m_{c}\big{[}m_{c}xy\chi^{\prime}(4x^{2}-3xy+4y^{2})-8m_{s}\chi(2x^{4}-x^{3}y-xy^{3}+2y^{4})\big{]}$
$\displaystyle-$
$\displaystyle\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{36864\sqrt{6}\pi^{4}\chi^{3}\chi^{\prime
4}}(1+5\beta)\big{[}-(Q^{2}+s-s^{\prime})x^{2}y\chi+2m_{c}m_{s}\chi^{\prime
2}\big{(}9x^{2}+9(y-1)^{2}+x(17y-18)\big{)}\Big{]}$ $\displaystyle-$
$\displaystyle\frac{m_{0}^{2}}{1536\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}\big{[}2m_{s}\big{(}\langle\bar{s}s\rangle(1+5\beta)-3(\beta-1)\langle\bar{u}u\rangle\big{)}xy+3m_{c}\big{(}\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}\chi^{\prime
2}$ $\displaystyle+$ $\displaystyle
3(\beta-1)\langle\bar{d}d\rangle\big{(}-2m_{s}xy+m_{c}\chi^{\prime
2}\big{)}\big{]}+\frac{1}{5184\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}(1+5\beta)g_{s}^{2}(\langle\bar{d}d\rangle^{2}+\langle\bar{s}s\rangle^{2}+\langle\bar{u}u\rangle^{2})xy$
$\displaystyle+$ $\displaystyle\frac{1}{48\sqrt{6}\pi^{2}\chi\chi^{\prime
4}}\big{[}(\beta-1)\langle\bar{s}s\rangle\langle\bar{u}u\rangle+\langle\bar{d}d\rangle\big{(}(\beta-1)\langle\bar{s}s\rangle+\langle\bar{u}u\rangle+5\beta\langle\bar{u}u\rangle\big{)}\big{]}xy\Bigg{\\}}\Theta[L(s,s^{\prime},Q^{2},x,y)]$

where

$\displaystyle\chi$ $\displaystyle=$ $\displaystyle(x+y-1),$
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle(x+y),$
$\displaystyle L(s,s^{\prime},Q^{2},x,y)$ $\displaystyle=$
$\displaystyle\frac{-Q^{2}xy-s^{\prime}(x+y-1)(x+y)-m_{c}^{2}(x+y)^{2}}{(x+y)^{2}},$
(16)

with $\Theta[\ldots]$ being the unit-step function.

Completing the calculations of both representations, we match the results
considering the coefficients of the same Lorentz structures. This step gives
two results both containing $g_{1}$ and $g_{2}$. Solving these two coupled
expressions together, we obtain the sum rules giving the considered coupling
constants, $g_{1}$ and $g_{2}$, analytically as

$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{cs}}^{2}}{M^{2}}}e^{\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{m_{J/\psi}(m_{J/\psi}^{2}+Q^{2})\left[(m_{P_{cs}}-m_{\Lambda})\tilde{\Pi}_{1}+\tilde{\Pi}_{2}\right]}{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}(m_{\Lambda}^{2}+m_{J/\psi}^{2}-m_{P_{cs}}^{2})}$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{cs}}^{2}}{M^{2}}}e^{\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{(m_{P_{cs}}+m_{\Lambda})(m_{J/\psi}^{2}+Q^{2})\left[m_{J/\psi}^{2}\tilde{\Pi}_{1}+(m_{P_{cs}}+m_{\Lambda})\tilde{\Pi}_{2}\right]}{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}m_{J/\psi}(m_{\Lambda}^{2}+m_{J/\psi}^{2}-m_{P_{cs}}^{2})},$
(17)

where $\tilde{\Pi}_{i}$ is the Borel transformed form of the $\Pi_{i}$
function. As is seen from the results for computations of these coupling
constants, we are in need of some input parameters. These parameters are
collected in Table 1. In the calculations, the masses of light $u$ and $d$
quarks are taken as zero.

Parameters | Values
---|---
$m_{s}$ |  $93^{+11}_{-5}~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{c}$ |  $(1.27\pm 0.02)~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{P_{cs}}$ |  $(4.47\pm 0.11)~{}\mathrm{GeV}$ Wang:2020eep
$m_{J/\psi}$ |  $(3096.900\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{\Lambda}$ |  $(1115.683\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$\lambda_{P_{cs}}$ |  $(1.86\pm 0.31)\times 10^{-3}~{}\mathrm{GeV}^{6}$ Wang:2020eep
$\lambda_{\Lambda}$ |  $(0.013\pm 0.02)~{}\mathrm{GeV}^{3}$ Aliev:2002ra
$f_{J/\psi}$ |  $(481\pm 36)~{}\mathrm{MeV}$ Veliev:2011kq
$\langle\bar{q}q\rangle$ |  $(-0.24\pm 0.01)^{3}$ $\mathrm{GeV}^{3}$ Belyaev:1982sa
$\langle\bar{s}s\rangle$ |  $0.8\langle\bar{q}q\rangle$ Belyaev:1982sa
$m_{0}^{2}$ |  $(0.8\pm 0.1)$ $\mathrm{GeV}^{2}$ Belyaev:1982sa
$\langle\overline{q}g_{s}\sigma Gq\rangle$ | $m_{0}^{2}\langle\bar{q}q\rangle$
$\langle g_{s}^{2}G^{2}\rangle$ |  $4\pi^{2}(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
Table 1: Some input parameters entering the calculations.

Besides the given parameters in Table 1, there are five additional parameters
which are the threshold parameters $s_{0}$ and $s^{\prime}_{0}$, Borel
parameters, $M^{2}$ and $M^{\prime 2}$ and the mixing parameter $\beta$ which
is coming from the interpolating current of the $\Lambda$ baryon. These
parameters are determined from the analyses of the results imposing the
standard criteria of the method such as weak dependence of the results on the
auxiliary parameters, pole dominance and convergence of operator product
expansion (OPE). Considering these conditions, the threshold parameters are
fixed as follows:

$\displaystyle 21.0\,\,\mathrm{GeV}^{2}$ $\displaystyle\leq$ $\displaystyle
s_{0}\leq 23.0\,\,\mathrm{GeV}^{2},$ $\displaystyle 1.7\,\,\mathrm{GeV}^{2}$
$\displaystyle\leq$ $\displaystyle s^{\prime}_{0}\leq
2.3\,\,\mathrm{GeV}^{2}.$ (18)

The upper limits of Borel parameters are determined by imposing the condition
of pole dominance for the selected working regions of continuum thresholds. To
this end, we consider the following ratio using the continuum subtracted and
Borel transformed invariant amplitude
$\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime 2},\beta)$ obtained from the QCD
side:

$\displaystyle PC=\frac{\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime
2},\beta)}{\Pi_{i}(\infty,\infty,M^{2},M^{\prime 2},\beta)},$ (19)

where $PC$ denotes the pole contribution and $i$ stands for the selected
structures. To fix the upper limit of the Borel parameters we impose this
ratio to be larger or at least equal to $20\%$, which is typical in the
analyses of the exotic states. For the calculations of their lower limits, the
convergence of the series of OPE is considered: the dominance of perturbative
part over the nonperturbative ones and ”the higher the dimension of the
nonperturbative operator, the lower its contribution”. To extract the lower
limit, using this criteria, we fix the ratio of the higher dimensional term,
that is the term having dimension 6 in the QCD side, to the whole result as
follows:

$\displaystyle R(M^{2},M^{\prime
2})=\frac{\Pi_{i}^{6}(s_{0},s^{\prime}_{0},M^{2},M^{\prime
2},\beta)}{\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime 2},\beta)},$ (20)

and keep this ratio as $R(M_{min}^{2},M^{\prime}{}^{2}_{min})=0.02$ to certify
the convergence of the OPE. With these conditions, the Borel parameters are
fixed as

$\displaystyle 5.0\ \mathrm{GeV}^{2}\leq M^{2}$ $\displaystyle\leq$
$\displaystyle 7.0\ \mathrm{GeV}^{2},$ $\displaystyle 1.4\
\mathrm{GeV}^{2}\leq M^{\prime 2}$ $\displaystyle\leq$ $\displaystyle 2.6\
\mathrm{GeV}^{2}.$ (21)

As the final parameter, we determine the working intervals of $\beta$ from the
analyses by considering a parametric plot of the results as functions of
$\cos\theta$ where $\beta=\tan\theta$. We select the regions that show least
variations with respect to the changes in $\cos\theta$, which read

$\displaystyle-1\leq\cos\theta\leq-0.5~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}0.5\leq\cos\theta\leq
1.$ (22)

Our analyses show that the physical quantities show weak dependence on the
auxiliary parameters in the above windows for $s_{0}$ and $s^{\prime}_{0}$,
$M^{2}$, $M^{\prime 2}$ and $\cos\theta$. To depict the dependence of the
results of coupling constants on the auxiliary parameters, we plot the Figures
1 and 2 for $g_{1}$ and $g_{2}$ at $Q^{2}=0$. From these figures and the
numerical values we see a good stability of the results with respect to the
Borel parameters in their working window. However, the results show some weak
dependencies on the continuum thresholds in their working intervals, which
remain inside the limits allowed by the method. The variations with respect to
the auxiliary parameters appear as the main sources of the uncertainties in
the numerical results.

Figure 1: Left: The variation of the strong coupling constant $g_{1}(Q^{2}=0)$
as a function of threshold parameters $s_{0}$ and $s_{0}^{\prime}$ at the
central values of the Borel parameters $M^{2}$ and $M^{\prime 2}$ and the
parameter $\beta$. Right: The variation of the strong coupling constant
$g_{1}(Q^{2}=0)$ as a function of Borel parameters $M^{2}$ and $M^{\prime 2}$
at the central values of the threshold parameters $s_{0}$ and $s_{0}^{\prime}$
and the parameter $\beta$.

Figure 2: Left: The variation of the strong coupling constant $g_{2}(Q^{2}=0)$
as a function of threshold parameters $s_{0}$ and $s_{0}^{\prime}$ at the
central values of the Borel parameters $M^{2}$ and $M^{\prime 2}$ and the
parameter $\beta$. Right: The variation of the strong coupling constant
$g_{2}(Q^{2}=0)$ as a function of Borel parameters $M^{2}$ and $M^{\prime 2}$
at the central values of the threshold parameters $s_{0}$ and $s_{0}^{\prime}$
and the parameter $\beta$.

Using the given input parameters in Table 1 and the determined windows for
auxiliary parameters, we calculate the strong coupling constants for the
considered decay channel. The following fit functions represent the
$Q^{2}$-behavior of the strong coupling form factors:

$\displaystyle g_{i}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{0}e^{c_{1}\frac{Q^{2}}{m_{P_{cs}}^{2}}+c_{2}(\frac{Q^{2}}{m_{P_{cs}}^{2}})^{2}}.$
(23)

with $g_{0}$, $c_{1}$ and $c_{2}$ being the fit parameters that take the
values given in Table 2.

Coupling Constant | $g_{0}$ | $c_{1}$ | $c_{2}$
---|---|---|---
$g_{1}$ | $4.22\pm 0.51$ | $1.54$ | $1.16$
$g_{2}$ | $10.54\pm 1.26$ | $1.54$ | $1.16$
Table 2: : Parameters of the fit functions for coupling constants, $g_{1}$ and
$g_{2}$.

We, then, use the fit functions to determine the coupling constants at
$Q^{2}=-m_{J/\psi}^{2}$ as

$\displaystyle g_{1}=2.63\pm
0.31~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}~{}~{}g_{2}=5.25\pm
0.63,$ (24)

where the errors are due to the uncertainties present in the input parameters
entering the calculation and in the determinations of the auxiliary
parameters, as well.

Having determined the strong coupling constants, the next task is to compute
the corresponding width for $P_{cs}\rightarrow J/\psi\Lambda$ decay channel in
terms of the strong coupling constants and other related parameters. The
standard calculations lead to the width formula as

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{f(m_{P_{cs}},m_{J/\psi},m_{\Lambda})}{16\pi
m_{P_{cs}}^{2}}\Bigg{[}-\frac{2(m_{J/\psi}^{2}-(m_{\Lambda}+m_{P_{cs}})^{2})}{m_{J/\psi}^{2}(m_{\Lambda}+m_{P_{cs}})^{2}}\Big{(}g_{2}^{2}m_{J/\psi}^{2}(m_{J/\psi}^{2}+2(m_{\Lambda}-m_{P_{cs}})^{2})$
(25) $\displaystyle+$ $\displaystyle
6g_{1}g_{2}m_{J/\psi}^{2}(m_{\Lambda}-m_{P_{cs}})(m_{\Lambda}+m_{P_{cs}})+g_{1}^{2}(2m_{J/\psi}^{2}+(m_{\Lambda}-m_{P_{cs}})^{2})(m_{\Lambda}+m_{P_{cs}})^{2}\Big{)}\Bigg{]},$

where

$\displaystyle f(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{2x}\sqrt{x^{4}+y^{4}+z^{4}-2x^{2}y^{2}-2x^{2}z^{2}-2y^{2}z^{2}}.$
(26)

Using the values of the strong coupling constants, we compute the width for
the considered channel to be

$\displaystyle\Gamma(P_{cs}\rightarrow J/\psi\Lambda)$ $\displaystyle=$
$\displaystyle\left(15.87\pm 3.11\right)~{}\mathrm{MeV}.$ (27)

##  III Summary and conclusion

The recently observed pentaquark state, the hidden-charmed strange
$P_{cs}(4459)^{0}$, added a new member to the pentaquark family. Its
experimentally observed mass and width were reported as $M=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV}$ and $\Gamma=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mathrm{MeV}$, respectively Aaij:2020gdg . However, its
quantum numbers, $J^{P}$, could not be determined as a result of insufficient
statistics in the experiment Aaij:2020gdg . Using the QCD sum rule method,
$P_{cs}(4459)^{0}$ state was studied both in the molecular form assigning its
quantum numbers as $J^{P}=\frac{1}{2}^{-}$ or $\frac{3}{2}^{-}$ Chen:2020uif
and in the diquark-diquark-antiquark form with quantum numbers
$\frac{1}{2}^{-}$ Wang:2020eep . Its mass was obtained in these studies and
compared to experimental data to shed light on its nature. Both of these
interpretations resulted in mass predictions consistent with the experimental
data creating a need for further investigations of this state, for instance
its width.

In this study, we investigated the strong $P_{cs}\rightarrow J/\psi\Lambda$
decay and obtained the strong coupling constants representing the amplitude of
this decay using the QCD sum rule method. To this end, we adopted an
interpolating current in the diquark-diquark-antiquark form for the
substructure of this particle. In the analysis, we considered the quantum
numbers of $P_{cs}(4459)^{0}$ state as $J^{P}=\frac{1}{2}^{-}$. The obtained
strong coupling constants were used in the determination of the corresponding
width, which is obtained as $\Gamma(P_{cs}\rightarrow
J/\psi\Lambda)=\left(15.87\pm 3.11\right)~{}\mathrm{MeV}$. Compared to the
experimental value, the obtained width is in good consistency with
experimental data, which favors the quantum numbers $J^{P}=\frac{1}{2}^{-}$
and compact pentaquark nature of diquark-diquark-antiquark form for
$P_{cs}(4459)^{0}$ state.

In Ref. Wang:2021itn , also, the authors have considered the molecular
interpretations for this state and concluded that it is either
$\bar{D}\Xi^{\prime}_{c}$ with $J^{P}=\frac{1}{2}^{-}$ and $I=0$ or
$\bar{D}\Xi^{*}_{c}$ with $J^{P}=\frac{3}{2}^{-}$ and $I=0$. In Ref.
Zhu:2021lhd  $P_{cs}(4459)$ state was interpreted as $\Xi_{c}\bar{D}^{*}$ with
$J^{P}=\frac{3}{2}^{-}$ without excluding the possibility of its being two-
pole structure $\Xi_{c}\bar{D}^{*}$ states with $J^{P}=\frac{1}{2}^{-}$ and
$J^{P}=\frac{3}{2}^{-}$. Another molecular interpretation was given in Ref.
Chen:2021tip  in which its two-body strong decay behaviors supported its being
$\Xi_{c}\bar{D}^{*}$ state with $I(J^{P})=0(\frac{3}{2}^{-})$.

All of the above-mentioned investigations indicate that the quantum numbers
and nature of the $P_{cs}(4459)^{0}$ state are still ambiguous and need
clarification not only from further theoretical studies of its various
properties but also from future experiments. Comparison of the theoretical
results on various parameters of this state with future experimental data will
shed light on the nature, quark-gluon organization and quantum numbers of this
state.

## ACKNOWLEDGEMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/99171.

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giving the conclusion for Dantzig. $Q.E.D.$

###  C.2 Large Sample Inference for Linear Effects

Proof of Lemma 4: Follows as in the proof of Lemma C1. $Q.E.D.$

Proof of Theorem 5: By Assumptions 8 and 9 and Theorems 1 and 3,

$\int[m(W,\hat{\gamma})-m(W,\gamma_{0})]^{2}F_{0}(dW)\overset{p}{\longrightarrow}0,\text{
}\|\hat{\gamma}_{L}-\gamma_{0}\|\overset{p}{\longrightarrow}0,\text{
}\|\hat{\alpha}_{L}-\alpha_{0}\|\overset{p}{\longrightarrow}0.$

Moreover, under Assumption 8 and Assumption 9 i)

$\sqrt{n}\|\hat{\alpha}_{L}-\alpha_{0}\|\|\hat{\gamma}-\gamma_{0}\|=O_{p}\left(n^{1/2}\left(\frac{\ln(p)}{n}\right)^{1/4}\sqrt{\ln(\ln(n))}n^{-d_{\gamma}}\right)\overset{p}{\longrightarrow}0.$

Likewise, under Assumption 8 and Assumption 9 ii)

$\sqrt{n}\|\hat{\alpha}_{L}-\alpha_{0}\|\|\hat{\gamma}-\gamma_{0}\|=O_{p}\left(n^{1/2}\left(\frac{\ln(p)}{n}\right)^{\xi/(1+2\xi)}\ln(\ln(n))n^{-d_{\gamma}}\right)\overset{p}{\longrightarrow}0.$

Then first conclusion then follows by Theorem 13 of Chernozhukov et al.
(2016).

The second conclusion follows similarly to the proof of Theorem 13 of
Chernozhukov et. al. (2016). Let $\psi_{i}=\psi_{0}(W_{i})$. Then for $i\in
I_{\ell},$

$\displaystyle(\hat{\psi}_{i}-\psi_{i})^{2}$ $\displaystyle\leq
C\left(\sum_{j=1}^{3}R_{ij}+R\right)=C\sum_{j=1}^{3}R_{ij}+o_{p}(1)$
$\displaystyle R_{i1}$
$\displaystyle=[m(W_{i},\hat{\gamma}_{\ell})-m(W_{i},\gamma_{0})]^{2},$
$\displaystyle R_{i2}$
$\displaystyle=\hat{\alpha}_{\ell}(X_{i})^{2}\\{\gamma_{0}(X_{i})-\hat{\gamma}(X_{i})\\}^{2},$
$\displaystyle R_{i3}$
$\displaystyle=\\{\hat{\alpha}_{\ell}(X_{i})-\alpha_{0}(X_{i})\\}^{2}\\{Y_{i}-\gamma_{0}(X_{i})\\}^{2},$
$\displaystyle R$ $\displaystyle=\\{\hat{\theta}-\theta_{0}\\}^{2}.$

The first conclusion implies $R\overset{p}{\longrightarrow}0$. Let $I_{-\ell}$
denote the observations not in $I_{\ell}.$

By Markov inequality,

$P\left(\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}>\delta\right)\leq\frac{E\left[\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}\right]}{\delta}.$

Write

$E\left[\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}\right]\leq
E\left[\frac{C}{n}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}\sum_{j=1}^{3}R_{ij}\right]+o_{p}(1)=C\sum_{\ell=1}^{L}\frac{n_{\ell}}{n}\sum_{j=1}^{3}EE[R_{ij}|I_{-\ell}]+o_{p}(1).$

Note that

$\max_{i\in
I_{\ell}}\left|\hat{\alpha}_{\ell}(X_{i})\right|\leq\left|\hat{\rho}_{L\ell}\right|_{1}\max_{i\in
I_{\ell}}\left|b(X_{i})\right|_{\infty}=O_{p}(1).$

Then for $i\in I_{\ell}$ by Assumptions 8 and 6

$\displaystyle E[R_{i1}|I{-\ell}]$
$\displaystyle=\int[m(W,\hat{\gamma}_{\ell})-m(W,\gamma_{0})]^{2}F_{0}(dW)\overset{p}{\longrightarrow}0$
$\displaystyle E[R_{i2}|I_{-\ell}]$ $\displaystyle\leq
O_{p}(1)\int[\hat{\gamma}_{\ell}(X)-\gamma_{0}(X)]^{2}F_{0}(dX)\overset{p}{\longrightarrow}0,\text{
}$ $\displaystyle E[R_{i3}|I_{-\ell}]$
$\displaystyle=E[\\{\hat{\alpha}_{\ell}(X_{i})-\alpha_{0}(X_{i})\\}^{2}\\{Y_{i}-\gamma_{0}(X_{i})\\}^{2}|X_{i},I_{-\ell}]$
$\displaystyle=E[\\{\hat{\alpha}_{\ell}(X_{i})-\alpha_{0}(X_{i})\\}^{2}Var(Y_{i}|X_{i})|I_{-\ell}]$
$\displaystyle\leq
C\|\hat{\alpha}_{\ell}-\alpha_{0}\|^{2}\overset{p}{\longrightarrow}0.$

In summary,

$\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}\overset{p}{\longrightarrow}0.$

Turning to the variance estimator, we have

$\displaystyle\hat{V}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\hat{\psi_{i}}^{2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i}+\psi_{i})^{2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}+2\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})\psi_{i}+\frac{1}{n}\sum_{i=1}^{n}\psi_{i}^{2};$

Furthermore by the Cauchy-Schwartz and Markov inequalities we have

$\left|\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})\psi_{i}\right|\leq\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\hat{\psi}_{i}-\psi_{i})^{2}}\sqrt{\frac{1}{n}\sum_{i=1}^{n}\psi_{i}^{2}}\overset{p}{\longrightarrow}0.$

Then $\hat{V}\overset{p}{\longrightarrow}V$ follows by the triangle inequality
and the law of large numbers. Q.E.D.

Proof of Theorem 6: By hypothesis i) for any measurable function $\gamma(X)$

$\displaystyle E[\gamma(1,Z)^{2}]$
$\displaystyle=E\left[\frac{D}{\pi_{0}(Z)}\gamma(1,Z)^{2}\right]=E\left[\frac{D}{\pi_{0}(Z)}\gamma(X)^{2}\right]\leq
CE[\gamma(X)^{2}],$ (C.4) $\displaystyle E[\gamma(0,Z)^{2}]$
$\displaystyle=E\left[\frac{1-D}{1-\pi_{0}(Z)}\gamma(0,Z)^{2}\right]=E\left[\frac{1-D}{1-\pi_{0}(Z)}\gamma(X)^{2}\right]\leq
CE[\gamma(X)^{2}].$

Then by $(a-b)^{2}\leq 2a^{2}+2b^{2}$ for any $a$ and $b$,

$E[m(W,\gamma)^{2}]=E[\\{\gamma(1,Z)-\gamma(0,Z)\\}^{2}]\leq
2E[\gamma(1,Z)^{2}]+2E[\gamma(0,Z)^{2}]\leq CE[\gamma(X)^{2}],$

so by i) and iii) Assumption 6 is satisfied. Note that ii) is Assumption 1.
Also by ii)

$\max_{1\leq j\leq p}|m(W,b_{j})-m(W,0)|\leq\max_{1\leq j\leq
p/2}|q_{j}(Z)|\leq C,$

so Assumption 7 is satisfied with $A(W)=1.$ In addition Assumption 8 is
satisfied by iv) and

$\displaystyle\int[m(w,\hat{\gamma})-m(w,\gamma_{0})]^{2}F_{W}(dw)$
$\displaystyle\quad\leq
2\int[\hat{\gamma}(1,z)-\gamma_{0}(1,z)]^{2}F_{W}(dw)+2\int[\hat{\gamma}(1,z)-\gamma_{0}(1,z)]^{2}F_{W}(dw)$
$\displaystyle\quad\leq C\int[\hat{\gamma}(x)-\gamma_{0}(x)]^{2}F_{W}(dw)$
$\displaystyle\quad=C\left\|\hat{\gamma}-\gamma_{0}\right\|^{2}\overset{p}{\longrightarrow}0,$

where the second inequality follows by eq. (C.4). Therefore Assumptions 1 and
6-9 are satisfied so the conclusion follows from Theorem 5. $Q.E.D.$

Proof of Theorem 7:  In this setting

$\int\gamma(x)d\mu(x)=m(W,\gamma)=E[m(W,\gamma)]=E[\alpha_{0}(X)\gamma(X)]=\int\alpha_{0}(x)\gamma(x)F_{X}(dx).$

By hypothesis i) for any measurable function $\gamma(X),$

$E[m(W,\gamma)^{2}]=m(W,\gamma)^{2}=\left\\{\int\alpha_{0}(x)\gamma(x)F_{X}(dx)\right\\}^{2}\leq
C[E\left|\gamma(X)\right|]^{2}\leq CE[\gamma(X)^{2}].$ (C.5)

Therefore by i) and iii) Assumption 6 is satisfied. Also ii) implies
Assumption 1. Also by ii)

$\max_{1\leq j\leq p}|m(W,b_{j})|\leq\max_{1\leq j\leq p}\left|\int
b_{j}(x)\alpha_{0}(x)F_{X}(dx)\right|\leq C\max_{1\leq j\leq p}\left|\int
b_{j}(x)F_{X}(dx)\right|\leq C,$

so Assumption 7 is satisfied with $A(W)=1.$ In addition Assumption 8 is
satisfied by iv) and

$\int[m(w,\hat{\gamma})-m(w,\gamma_{0})]^{2}F_{W}(dw)=[m(W,\hat{\gamma})-m(W,\gamma_{0})]^{2}=[m(W,\hat{\gamma}-\gamma)]^{2}\leq
C\left\|\hat{\gamma}-\gamma_{0}\right\|^{2}\overset{p}{\longrightarrow}0,$

where the inequality follows by eq. (C.5). Therefore Assumptions 1 and 6-9 are
satisfied so the conclusion follows from Theorem 5. $Q.E.D.$

Proof of Theorem 8:  By hypothesis i) for any measurable function $\gamma(X),$

$\displaystyle E[m(W,\gamma)]^{2}$
$\displaystyle=\int[\gamma(t(x))-\gamma(x)]^{2}dF(x)$ (C.6) $\displaystyle\leq
2\int[\gamma^{2}(t(x))+\gamma^{2}(x)]dF(x)$
$\displaystyle=2\int\gamma^{2}(x)\left(\frac{f_{t}(x)}{f(x)}-1\right)f(x)dx$
$\displaystyle\leq CE\gamma^{2}(X)$

Therefore by i) and iii) Assumption 6 is satisfied. Also ii) implies
Assumption 1. Also by ii)

$\max_{1\leq j\leq p}|m(W,b_{j})|\leq\max_{1\leq j\leq
p}|b_{j}(t(x))-b_{j}(x)|\leq\max_{1\leq j\leq
p}\left|b_{j}(t(x))\right|+\max_{1\leq j\leq p}\left|b_{j}(x)\right|\leq C,$

so Assumption 7 is satisfied with $A(W)=1.$ In addition Assumption 8 is
satisfied by iv) and

$\displaystyle\int[m(w,\hat{\gamma})-m(w,\gamma_{0})]^{2}F_{W}(dw)$
$\displaystyle\leq
C\left\|\hat{\gamma}-\gamma_{0}\right\|^{2}\overset{p}{\longrightarrow}0.$

where the inequality follows by eq. (C.6). Therefore Assumptions 1 and 6-9 are
satisfied so the conclusion follows from Theorem 5. $Q.E.D.$

Proof of Theorem 9: Assumption 6 is satisfied by i) and iii). Assumption 7 is
satisfied by v). Assumptions 1, 8, and 9 are satisfied by hypothesis, so the
conclusion follows by Theorem 5. $Q.E.D.$

###  C.3 GMM with Moment Functions Nonlinear in Regressions

Proof of Lemma 10: Define

$\bar{M}^{k}(\gamma,\theta)=(\bar{M}_{1}^{k}(\gamma,\theta),...,\bar{M}_{p}^{k}(\gamma,\theta))^{\prime},\text{
}\bar{M}_{j}^{k}(\gamma,\theta)=\int D^{k}(W,b_{j},\gamma,\theta)F(dW).$

For notational convenience we henceforth suppress the $k$ superscript and
assume that $\theta$ is included with $\gamma$, e.g. so that
$M=\bar{M}^{k}(\gamma_{0},\theta_{0})$, $\gamma_{0}=(\gamma_{0},\theta_{0})$,
and
$\tilde{\gamma}_{\ell,\ell^{\prime}}=(\tilde{\gamma}_{\ell,\ell^{\prime}},\tilde{\theta}_{\ell,\ell^{\prime}}).$
Let $\Gamma_{\ell,\ell^{\prime}}$ be the event that
$\|\tilde{\gamma}_{\ell,\ell^{\prime}}-\gamma_{0}\|\leq\varepsilon$ and note
that $\Pr(\Gamma_{\ell,\ell^{\prime}})\longrightarrow 1$ for each $\ell$ and
$\ell^{\prime}$. When $\Gamma_{\ell,\ell^{\prime}}$ occurs,

$\max_{j}|D(W_{i},b_{j},\tilde{\gamma}_{\ell,\ell^{\prime}})|\leq C$

by Assumption 11. Define

$T_{ij}(\gamma)=D(W_{i},b_{j},\gamma)-\bar{M}_{j}(\gamma),\text{ }(i\in
I_{\ell^{\prime}}),\text{
}U_{\ell^{\prime}j}(\gamma)=\dfrac{1}{n_{\ell^{\prime}}}\sum_{i\in
I_{\ell^{\prime}}}T_{ij}(\gamma).$

Note that for any constant $C^{\prime}$ and the event
$\mathcal{A=\\{}\max_{j}|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|\geq
C^{\prime}\varepsilon_{n}^{\ast}\\}$ where
$\varepsilon_{n}^{\ast}=\sqrt{\ln(p)/n}$

$\displaystyle\Pr(\mathcal{A})$
$\displaystyle=\Pr(\mathcal{A}|\Gamma_{\ell,\ell^{\prime}})\Pr(\Gamma_{\ell,\ell^{\prime}})+\Pr(\mathcal{A}|\Gamma_{\ell,\ell^{\prime}}^{c})\left[1-\Pr(\Gamma_{\ell,\ell^{\prime}})\right]$
(C.7)
$\displaystyle\leq\Pr(\max_{j}|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|\geq
C^{\prime}\varepsilon_{n}^{\ast}|\Gamma_{\ell,\ell^{\prime}})+1-\Pr(\Gamma_{\ell,\ell^{\prime}}).$

Also

$\Pr(\max_{j}|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|\geq
C^{\prime}\varepsilon_{n}^{\ast}|\Gamma_{\ell,\ell^{\prime}})\leq
p\cdot\max_{j}\Pr(|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|>C^{\prime}\varepsilon_{n}^{\ast}|\Gamma_{\ell,\ell^{\prime}}).$

Note that
$E[T_{ij}(\tilde{\gamma}_{\ell,\ell^{\prime}})|\tilde{\gamma}_{\ell,\ell^{\prime}}]=0$
for $i\in I_{\ell^{\prime}}.$ Also, conditional on the event
$\Gamma_{\ell,\ell^{\prime}},$

$|T_{ij}(\tilde{\gamma}_{\ell,\ell^{\prime}})|\leq 2C,\text{ }i\in
I_{\ell^{\prime}}.$

Define $K=2C/\ln 2$. By Hoeffding’s inequality (Vershynin, 2018) and the
independence of $(W_{i})_{i\in I_{\ell^{\prime}}}$ and
$\tilde{\gamma}_{\ell,\ell^{\prime}}$ there is a constant $c$ such that

$\displaystyle
p\cdot\max_{j}\Pr(|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|$
$\displaystyle>C^{\prime}\varepsilon_{n}^{\ast}|\Gamma_{\ell,\ell^{\prime}})$
$\displaystyle=p\cdot\max_{j}E[\Pr(|U_{\ell^{\prime}j}(\tilde{\gamma}_{\ell,\ell^{\prime}})|>C^{\prime}\varepsilon_{n}^{\ast}|\tilde{\gamma}_{\ell,\ell^{\prime}})|\Gamma_{\ell,\ell^{\prime}}]$
$\displaystyle\leq
2p\exp\left(-\dfrac{c(n_{\ell^{\prime}}C^{\prime}\varepsilon_{n}^{\ast})^{2}}{n_{\ell^{\prime}}K^{2}}\right)$
$\displaystyle\leq 2p\exp\left(-\dfrac{cC^{\prime 2}\ln(p)}{LK^{2}}\right)$
$\displaystyle\leq 2\exp\left(\ln(p)\left[1-\dfrac{cC^{\prime
2}}{LK^{2}}\right]\right)\longrightarrow 0,$

for any $C^{\prime}>K\sqrt{\frac{L}{c}}.$ Let
$U_{\ell^{\prime}}(\gamma)=(U_{\ell^{\prime}1}(\gamma),...,U_{\ell^{\prime}p}(\gamma))^{\prime}.$
It then follows from equation (C.7) that for large $C^{\prime}$,
$\Pr(|U_{\ell^{\prime}}(\tilde{\gamma}_{\ell,\ell^{\prime}})|_{\infty}\geq
C^{\prime}\varepsilon_{n}^{\ast})\longrightarrow 0.$ Therefore
$|U_{\ell^{\prime}}(\tilde{\gamma}_{\ell,\ell^{\prime}})|_{\infty}=O_{p}(\varepsilon_{n}^{\ast}).$

Next, for each $\ell$ it follows that
$n-n_{\ell}=\sum_{\ell^{\prime}\neq\ell}n_{\ell^{\prime}}$ and

$\left|\hat{M}_{\ell}-\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell}}\bar{M}(\tilde{\gamma}_{\ell,\ell^{\prime}})\right|_{\infty}=\left|\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell}}U_{\ell^{\prime}}(\tilde{\gamma}_{\ell,\ell^{\prime}})\right|_{\infty}\leq\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell}}|U_{\ell^{\prime}}(\tilde{\gamma}_{\ell,\ell^{\prime}})|_{\infty}=O_{p}(\varepsilon_{n}^{\ast}).$

Also by Assumption 11 ii) and $\Pr(\Gamma_{\ell,\ell^{\prime}})\longrightarrow
1$ for each $\ell$ and $\ell^{\prime},$

$\left|\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell}}\bar{M}(\tilde{\gamma}_{\ell,\ell^{\prime}})-M\right|_{\infty}=\left|\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell^{\prime}}}[\bar{M}(\tilde{\gamma}_{\ell,\ell^{\prime}})-M]\right|_{\infty}\leq
C\sum_{\ell^{\prime}\neq\ell}\frac{n_{\ell^{\prime}}}{n-n_{\ell^{\prime}}}\|\tilde{\gamma}_{\ell,\ell^{\prime}}-\gamma_{0}\|=O_{p}(n^{-d_{\gamma}}).$

The conclusion then follows by the triangle inequality. $Q.E.D.$

Proof of Theorem 11: We first the conditions of Theorems 16 and 17 of
Chernozhukov et al. (2016). Let $\phi(w,\gamma,\alpha)$ in Chernozhukov et al.
(2016) be $\alpha(x)[y-\gamma(x)].$ By Assumption 6 and Lemma 11 it follows
that

$\displaystyle\int[\phi(w,\hat{\gamma},\alpha_{0})-\phi(w,\gamma_{0},\alpha_{0})]^{2}F_{0}(dw)$
$\displaystyle=\int\alpha_{0}(x)^{2}[\hat{\gamma}(x)-\gamma_{0}(x)]^{2}F_{0}(dx)\leq
C\|\hat{\gamma}-\gamma_{0}\|^{2}\overset{p}{\longrightarrow}0.$
$\displaystyle\int[\phi(w,\gamma_{0},\hat{\alpha})-\phi(w,\gamma_{0},\alpha_{0})]^{2}F_{0}(dW)$
$\displaystyle=\int[\hat{\alpha}(x)-\alpha_{0}(x)]^{2}[y-\gamma_{0}(x)]^{2}F_{0}(dw)$
$\displaystyle=\int[\hat{\alpha}(x)-\alpha_{0}(x)]^{2}Var(Y|X=x)F_{0}(dx)$
$\displaystyle\leq
C\|\hat{\alpha}-\alpha_{0}\|^{2}\overset{p}{\longrightarrow}0.$

Also by Assumption 8 with $m(w,\gamma)$ equal to $g(w,\gamma,\theta_{0})$ we
have

$\int[g(w,\hat{\gamma},\theta_{0})-g(w,\gamma_{0},\theta_{0})]^{2}F_{0}(dw)\overset{p}{\longrightarrow}0,$

so all the conditions of Assumption 4 of Chernozhukov et al. (2016) are
satisfied.

By Assumptions 8 and 11 it follows from Lemma 10 that Assumption 2 is
satisfied with $\varepsilon_{n}^{M}=n^{-d_{\gamma}}.$ Note that if Assumption
12 i) is satsified then

$\sqrt{n}r_{L}^{1/2}n^{-d_{\gamma}}=O(\sqrt{n}\sqrt{\ln(\ln(n))}n^{-3d_{\gamma}/2})=n^{(1-3d_{\gamma})/2}\sqrt{\ln(\ln(n))}\longrightarrow
0.$

Then it follows from Theorem 1 and Assumption 8 that

$\sqrt{n}\|\hat{\alpha}_{\ell}-\alpha_{0}\|\|\hat{\gamma}_{\ell}-\gamma_{0}\|=O_{p}(\sqrt{n}r_{L}^{1/2}n^{-d_{\gamma}})\overset{p}{\longrightarrow}0.$

Also if Assumption 12 ii) is satisfied then for
$\varepsilon_{n}=n^{-d_{\gamma}}$ then

$\sqrt{n}\varepsilon_{n}^{-1/(1+2\xi)}r_{L}n^{-d_{\gamma}}=O(\sqrt{n}\varepsilon_{n}^{2\xi/(1+2\xi)}\ln(\ln(n))n^{-d_{\gamma}})=O(n^{1/2-d_{\gamma}(1+4\xi)/(1+2\xi)}\ln(\ln(n)))\longrightarrow
0.$

Then it follows from Theorem 3 and Assumption 8 that

$\sqrt{n}\|\hat{\alpha}_{\ell}-\alpha_{0}\|\|\hat{\gamma}_{\ell}-\gamma_{0}\|=O_{p}(\sqrt{n}\varepsilon_{n}^{-1/(1+2\xi)}r_{L}n^{-d_{\gamma}})\overset{p}{\longrightarrow}0.$

Then by Cauchy-Schwarz inequality,

$\displaystyle\sqrt{n}\int|\phi(w,\hat{\gamma}_{\ell},\hat{\alpha}_{\ell})-\phi(w,\gamma_{0},\hat{\alpha}_{\ell})-\phi(w,\hat{\gamma}_{\ell},\alpha_{0})+\phi(w,\gamma_{0},\alpha_{0})|F_{0}(dw)$
$\displaystyle=\sqrt{n}\int|(\hat{\alpha}_{\ell}(x)-\alpha_{0}(x))\cdot(\hat{\gamma}_{\ell}(x)-\gamma_{0}(x))|F_{0}(dx)$
$\displaystyle\leq\sqrt{n}\|\hat{\alpha}_{\ell}-\alpha_{0}\|\|\hat{\gamma}_{\ell}-\gamma_{0}\|,$

so that Assumption 5 of Chernozhukov et al. (2016) is satisfied.

Also, by iterated expectations and Assumption B1,

$\displaystyle\sqrt{n}\int\psi(w,\hat{\gamma}_{\ell},{\alpha}_{0})F_{0}(dw)$
$\displaystyle=\sqrt{n}\int[g(w,\hat{\gamma}_{\ell},\theta_{0})+\alpha_{0}(x)\\{y-\hat{\gamma}_{\ell}(x)\\}]F_{0}(dw)$
$\displaystyle=\sqrt{n}\int[g(w,\hat{\gamma}_{\ell},\theta_{0})-g(w,\gamma_{0},\theta_{0})+\alpha_{0}(x)\\{y-\hat{\gamma}_{\ell}(x)\\}]F_{0}(dw)$
$\displaystyle=\sqrt{n}\int[g(w,\hat{\gamma}_{\ell},\theta_{0})-g(w,\gamma_{0},\theta_{0})+\alpha_{0}(x)\\{\gamma_{0}(x)-\hat{\gamma}_{\ell}(x)\\}]F_{0}(dw)$
$\displaystyle=\sqrt{n}\int[g(w,\hat{\gamma}_{\ell},\theta_{0})-g(w,\gamma_{0},\theta_{0})-D(w,\hat{\gamma}_{\ell}-\gamma_{0},\gamma_{0})]F_{0}(dw)$
$\displaystyle\leq C\sqrt{n}\|\hat{\gamma}_{\ell}-\gamma_{0}\|^{2}$
$\displaystyle=C\sqrt{n}O_{p}(n^{-2d_{\gamma}})\overset{p}{\longrightarrow}0.$

Furthermore,

$\sqrt{n}\left|\int\hat{\alpha}_{\ell}(x)\\{y-\gamma_{0}(x)\\}]F_{0}(dw)\right|=0.$

Therefore Assumption 6 of Chernozhukov et al. (2016) is satisfied.

Assumption 7 of Chernozhukov et al. (2016) follows from Assumptions B2 and B3.
It also follows similarly to the proof of Theorem 5 that there is $C$ with
probability approach one

$\int\left|\\{\hat{\alpha}_{\ell}(x)-\alpha_{0}(x)\\}\\{\hat{\gamma}_{\ell}(x)-\gamma_{0}(x)\\}\right|_{2}^{2}F_{0}(dx)\leq
C\int\left|\hat{\gamma}_{\ell}(x)-\gamma_{0}(x)\right|_{2}^{2}F_{0}(dx)\overset{p}{\longrightarrow}0.$

The conclusion then follows from Theorems 16 and 17 of Chernozhukov et al.
(2016). $Q.E.D.$

Proof of Theorem 12: We proceed by verifying the conditions of Theorem 11.
Note condition i) of Theorem 11 is satisfied by hypothesis. Assumption 1 is
satisfied by hypothesis.

For verifying Assumption 6, note that $Var(Y|X)$ is bounded by hypothesis.
Also, $D/\pi_{0}(Z)-(1-D)/[1-\pi_{0}(Z)]$ is bounded by $\pi_{0}(Z)\in[C,1-C]$
and $\phi(\varepsilon)$ and $H(v,z)$ are bounded, so that $\alpha_{0}(x)$ is
bounded. In addition $g(w,\gamma,\theta)$ is bounded by $H(v,z)$ and
$\Phi(\varepsilon)$ bounded.

For the remainder of the proof we lighten notation by defining

$r(z,\gamma)=\gamma(1,z)-\gamma(0,z).$

From $\pi_{0}(Z)\in[C,1-C]$ it follows similarly to proof of Theorem 6 that

$E[r(Z,\gamma)^{2}]\leq C\left\|\gamma\right\|^{2}.$ (C.8)

We use this result in the remainder of the proof.

For Assumption 8, note that by boundedness of $H(v,z)$ and the pdf
$\phi(\varepsilon)$ we have

$\int|g^{k}(w,\hat{\gamma}_{\ell},\theta_{0})-g^{k}(w,\gamma_{0},\theta_{0})|^{2}F(dw)\leq
C\int|r(z,\hat{\gamma}_{\ell})-r(z,\gamma_{0})|^{2}F(dw)\leq\left\|\hat{\gamma}_{\ell}-\gamma_{0}\right\|^{2}\overset{p}{\longrightarrow}0,$

where the second inequality holds by equation (C.8). The other condition of
Assumption 8 holds by hypothesis.

For Assumption 11 note that $\left|D^{k}(w,b_{j},\gamma,\theta)\right|$ is
bounded uniformly in $j$ and $k$ by $H^{k}(v,z)$ bounded uniformly in $k$,
$\phi(\varepsilon)$ bounded, and $b_{j}(x)$ bounded uniformly in $j$ and $x$.
Also, for the same reasons and
$\phi_{\varepsilon}(\varepsilon)=d\phi(\varepsilon)/d\varepsilon$ bounded

$\displaystyle|E[D^{k}(W,b_{j},\gamma,\theta)-D(W,b_{j},\gamma_{0},\theta_{0})]|$
$\displaystyle\quad\leq CE[\left|V^{\prime}\beta+\delta
r(Z,\gamma)-V^{\prime}\beta_{0}-\delta_{0}r(Z,\gamma_{0})\right|]$
$\displaystyle\quad\leq
C(\left|\beta-\beta_{0}\right|_{2}+CE[\left|r(Z,\gamma)-r(Z,\gamma_{0})\right|]+C\left|\delta-\delta_{0}\right|E[\left|r(Z,\gamma_{0})\right|]$
$\displaystyle\quad\leq
C\left|\theta-\theta_{0}\right|_{2}+C\sqrt{E[\left|r(Z,\gamma)-r(Z,\gamma_{0})\right|^{2}]}$
$\displaystyle\quad\leq
C\left|\theta-\theta_{0}\right|_{2}+C\left\|\gamma-\gamma_{0}\right\|,$

where the third inequality holds by equation (C.8). Thus Assumption 11 is
satisfied.

Assumption 12 is satisfied by hypothesis.

To verify Assumption B1, let $r$ and $r_{0}$ be scalars, and note that by
Taylor expansion with LaGrange remainder and by
$\phi_{\varepsilon}(\varepsilon)=d\phi(\varepsilon)/d\varepsilon$ bounded,

$\Phi(v^{\prime}\beta_{0}+\delta_{0}r)=\Phi(v^{\prime}\beta_{0}+\delta_{0}r_{0})+\phi(v^{\prime}\beta_{0}+\delta_{0}r_{0})\delta_{0}(r-r_{0})+R(v,r,r_{0}),\text{
}\left|R(v,r,r_{0})\right|\leq C(r-r_{0})^{2}.$

Then by $H(v,z)$ bounded it follows that for $r=\gamma(1,z)-\gamma(0,z)$ and
$r_{0}=\gamma_{0}(1,z)-\gamma_{0}(0,z)$

$\left|g^{k}(w,\gamma,\theta_{0})-g^{k}(w,\gamma_{0},\theta_{0})-D^{k}(w,\gamma-\gamma_{0},\gamma_{0},\theta_{0})\right|\leq
C\\{r(z,\gamma)-r(z,\gamma_{0})\\}^{2}.$

Therefore by equation (C.8) we have

$\left|E[g^{k}(W,\gamma,\theta_{0})-g^{k}(W,\gamma_{0},\theta_{0})-D^{k}(W,\gamma-\gamma_{0},\gamma_{0},\theta_{0})]\right|\leq
CE[\\{r(Z,\gamma)-r(Z,\gamma_{0})\\}^{2}]\leq
C\left\|\gamma-\gamma_{0}\right\|^{2}.$

where the last inequality follows from $\pi_{0}(Z)\in[C,1-C]$ similarly to
proof of Theorem 6. Therefore, Assumption B1 is satisfied.

To show Assumption B2, note that by $\Phi(\varepsilon)$ twice differentiable
and the chain rule it follows that $g^{k}(W,\gamma,\theta)$ is twice
differentiable everywhere, implying i) of Assumption B2. By condition v)
$\delta_{0}\neq 0$ so that for $\varepsilon$ small enough
$\left|\theta-\theta_{0}\right|<\varepsilon$ implies $\delta^{2}\geq C>0.$
Then by differentiating twice with respect to $\theta$ it follows by
$H^{k}(V,Z)$, $V$, and $V^{\prime}\beta$ bounded uniformly in $\beta$ in a
neighborhood of $\beta_{0}$ that there is $C$ big enough such that for

$\displaystyle\left|\frac{\partial
g^{k}(W,\gamma,\theta)}{\partial\theta\partial\theta^{\prime}}\right|_{F}$
$\displaystyle\leq C\left|\phi_{\varepsilon}(V^{\prime}\beta+\delta
r(Z,\gamma))\right|(1+r(Z,\gamma))^{2}$ $\displaystyle\leq
C\left|\phi_{\varepsilon}(V^{\prime}\beta+\delta
r(Z,\gamma))\right|(1+r(Z,\gamma)^{2})$
$\displaystyle=C\left|\phi_{\varepsilon}(V^{\prime}\beta+\delta
r(Z,\gamma))\right|(1+\delta^{-2}\\{\delta r(Z,\gamma)\\}^{2})$
$\displaystyle\leq C\left|\phi_{\varepsilon}(V^{\prime}\beta+\delta
r(Z,\gamma))\right|(1+\\{\delta r(Z,\gamma)\\}^{2})$ $\displaystyle\leq
C\max_{\left|a\right|\leq C}\left|\phi_{\varepsilon}(a+b)\right|(1+b^{2})$
$\displaystyle\leq C.$

Thus Assumption B2 ii) is satisfied with $d(W)=C.$

Finally we verify Assumption B3. By $H(v,z)$ bounded and by
$\phi(\varepsilon)$ bounded with bounded derivative

$\displaystyle\left|\frac{\partial
g^{k}(W,\hat{\gamma},\theta_{0})}{\partial\theta}-\frac{\partial
g^{k}(W,\gamma_{0},\theta_{0})}{\partial\theta}\right|_{2}$
$\displaystyle\quad\leq
C\left|\phi(V^{\prime}\beta_{0}+\delta_{0}r(Z,\hat{\gamma}))-\phi(V^{\prime}\beta_{0}+\delta_{0}r(Z,\gamma_{0}))\right|$
$\displaystyle\quad\quad+C\left|\phi(V^{\prime}\beta_{0}+\delta_{0}r(Z,\hat{\gamma}))r(Z,\hat{\gamma})-\phi(V^{\prime}\beta_{0}+\delta_{0}r(Z,\gamma_{0}))r(Z,\gamma_{0})\right|$
$\displaystyle\quad\leq
C\left|r(Z,\hat{\gamma})-r(Z,\gamma_{0})\right|+C\phi(V^{\prime}\beta_{0}+\delta_{0}r(Z,\hat{\gamma}))\left|r(Z,\hat{\gamma})-r(Z,\gamma_{0})\right|+C\left|r(Z,\gamma_{0})\right|\left|r(Z,\hat{\gamma})-r(Z,\gamma_{0})\right|$
$\displaystyle\quad\leq
C(1+\left|r(Z,\gamma_{0})\right|)\left|r(Z,\hat{\gamma})-r(Z,\gamma_{0})\right|.$

Note that similarly to the proof of Theorem 6,
$\int\\{r(z,\hat{\gamma}_{\ell})-r(z,\gamma_{0})\\}^{2}F(dw)\leq
C\left\|\hat{\gamma}_{\ell}-\gamma_{0}\right\|^{2}\overset{p}{\longrightarrow}0.$
Therefore we have

$\displaystyle\int\left|\frac{\partial
g^{k}(w,\hat{\gamma},\theta_{0})}{\partial\theta}-\frac{\partial
g^{k}(w,\gamma_{0},\theta_{0})}{\partial\theta}\right|_{2}F(dw)$
$\displaystyle\leq
C\sqrt{\int\\{1+r(z,\gamma_{0})^{2}\\}F(dw)}\sqrt{\int\\{r(z,\hat{\gamma}_{\ell})-r(z,\gamma_{0})\\}^{2}F(dw)}\overset{p}{\longrightarrow}0,$

We have now verified all the conditions of Theorem 11, so the conclusion
follows by Theorem 11. $Q.E.D.$

###  C.4 Regression Decomposition and ATT

Proof of Theorem 13: Note first that for any $\gamma(D,Z)$ it follows as in
the proof of Theorem 6 that by $\Pr(D=1|Z)<1-C,$

$E[D\gamma(0,Z)^{2}]\leq E[\gamma(0,Z)^{2}]\leq CE[\gamma(X)^{2}].$

Also note that

$E[D\gamma(0,Z)]=E[\pi_{0}(Z)\gamma(0,Z)]=E\left[\pi_{0}(Z)\frac{1-D}{1-\pi_{0}(X)}\gamma(0,Z)\right]=E[\alpha_{0}(X)\gamma(X)].$

The remainder of the proof follows anagously to the proof of Theorem 6.
$Q.E.D.$

##  Appendix D Delta Method for Smooth Transformations of Linear Effects

###  D.1 Regression Decomposition and ATT

Since ATT $\theta_{0}^{*}$ is a deterministic mapping of
$\tilde{\theta}_{0}:=(\theta_{0},\mathbb{E}[DY],\mathbb{E}[D])^{\prime},$ we
obtain the asymptotic variance $V^{*}$ of $\theta^{*}_{0}$ from the asymptotic
variance $\tilde{V}$ of $\tilde{\theta}_{0}$ using delta method. Specifically,

$V^{*}=H\tilde{V}H^{\prime}$

where

$H=\frac{\partial\theta^{*}_{0}}{\partial\tilde{\theta}_{0}}=\begin{bmatrix}-\frac{1}{\mathbb{E}[D]}&\frac{1}{\mathbb{E}[D]}&\frac{\theta_{0}-\mathbb{E}[DY]}{[\mathbb{E}[D]]^{2}}\end{bmatrix}$

and

$\displaystyle\tilde{V}=\begin{bmatrix}\mathbb{E}[\psi_{0}(W)]^{2}&\mathbb{E}[\psi_{0}(W)DY]&\mathbb{E}[\psi_{0}(W)D]\\\
\cdot&\mathbb{E}[DY]^{2}-\left\\{\mathbb{E}[DY]\right\\}^{2}&\left\\{1-\mathbb{E}[D]\right\\}\mathbb{E}[DY]\\\
\cdot&\cdot&\left\\{1-\mathbb{E}[D]\right\\}\mathbb{E}[D]\end{bmatrix}.$

We estimate the asymptotic variance $V^{*}$ using the empirical analogue
$\hat{V}^{*}$, where $(D,Y)$ are replaced by $(D_{i},Y_{i})$ and $\psi_{0}(W)$
is replaced by

$\displaystyle\hat{\psi}_{i}$
$\displaystyle=D_{i}\hat{\gamma}_{\ell}(0,Z_{i})-\hat{\theta}+\hat{\alpha}_{\ell}(D_{i},Z_{i})[Y_{i}-\hat{\gamma}_{\ell}(D_{i},Z_{i})],\quad
i\in I_{\ell}$

in both $H$ and $\tilde{V}$. For example, the estimator for the first
component of $\tilde{V}$ is

$\frac{1}{n}\sum_{i=1}^{n}\left(\hat{\psi}_{i}\right)^{2}.$

###  D.2 Panel Average Derivative and Demand Elasticities

Since own-price elasticity $\theta^{*}_{0}$ is a deterministic mapping of
$\tilde{\theta}_{0}:=(\theta_{0},\mathbb{E}[Y_{it}])^{\prime}$, we obtain the
asymptotic variance $V^{*}$ of $\theta^{*}_{0}$ from the asymptotic variance
$\tilde{V}$ of $\tilde{\theta}_{0}$ using delta method. Specifically,

$V^{*}=H\tilde{V}H^{\prime}$

where

$H=\frac{\partial\theta^{*}_{0}}{\partial\tilde{\theta}_{0}}=\begin{bmatrix}\frac{1}{\mathbb{E}[Y_{it}]}&\frac{-\theta_{0}}{[\mathbb{E}[Y_{it}]]^{2}}\end{bmatrix}$

and

$\displaystyle\tilde{V}=\begin{bmatrix}\mathbb{E}[\psi_{0}(W_{it})]^{2}&\mathbb{E}[\psi_{0}(W_{it})Y_{it}]\\\
\cdot&\mathbb{E}[Y_{it}]^{2}-\left\\{\mathbb{E}[Y_{it}]\right\\}^{2}\end{bmatrix}.$

We estimate the asymptotic variance $V^{*}$ using the empirical analogue
$\hat{V}^{*}$, where $\psi_{0}(W_{it})$ is replaced by

$\displaystyle\hat{\psi}_{it}$
$\displaystyle=\frac{\partial\hat{\gamma}_{\ell}(\tilde{X}_{i})}{\partial
d}-\hat{\theta}+\hat{\alpha}^{*}_{\ell}(\tilde{X}_{i})[Y_{it}-\hat{\gamma}_{\ell}(\tilde{X}_{i})],\quad
i\in I_{\ell}.$

The covariance estimator recognizes that household $i$’s observations form a
cluster $T_{i}$. For example, the estimator for the first component of
$\tilde{V}$ is

$\frac{1}{\sum_{i=1}^{n}T_{i}}\sum_{i=1}^{n}\sum_{t\in T_{i}}\sum_{s\in
T_{i}}\hat{\psi}_{it}\hat{\psi}_{is}.$

More generally, we may consider estimating not only own price elasticity but
also income elasticity and cross price elasticity. The same arguments go
through with light modification.

Concatenate the derivatives as

$\theta_{0}=\begin{bmatrix}(\theta_{0})^{\text{income}}\\\
(\theta_{0})^{\text{own}}\\\ (\theta_{0})^{\text{cross}}\\\
\end{bmatrix}=\begin{bmatrix}E\left[\frac{\partial\gamma_{0}(\tilde{X}_{i})}{\text{log
income}}\right]\\\ E\left[\frac{\partial\gamma_{0}(\tilde{X}_{i})}{\text{log
own price}}\right]\\\
E\left[\frac{\partial\gamma_{0}(\tilde{X}_{i})}{\text{log cross
price}}\right]\\\ \end{bmatrix}$

where the first and second components are scalars and the third component is a
vector.

The elasticities are a smooth transform thereof. By arguments in Chernozhukov,
Hausman, and Newey (2019)

$\theta^{*}_{0}=\begin{bmatrix}(\theta_{0}^{*})^{\text{income}}\\\
(\theta_{0}^{*})^{\text{own}}\\\ (\theta_{0}^{*})^{\text{cross}}\\\
\end{bmatrix}=\begin{bmatrix}\frac{(\theta_{0})^{\text{income}}}{\mathbb{E}[Y_{it}]}-1\\\
\frac{(\theta_{0})^{\text{own}}}{\mathbb{E}[Y_{it}]}-1\\\
\frac{(\theta_{0})^{\text{cross}}}{\mathbb{E}[Y_{it}]}\end{bmatrix}.$

Likewise the delta method argument goes through. Elasticites $\theta_{0}$ are
a deterministic mapping of
$\tilde{\theta}_{0}=((\theta_{0}^{*})^{\prime},\mathbb{E}[Y_{it}])^{\prime}$.
We obtain the asymptotic variance $V^{*}$ of $\theta^{*}_{0}$ from the
asymptotic variance $\tilde{V}$ of $\tilde{\theta}_{0}$ using delta method.
Specifically,

$V^{*}=H\tilde{V}H^{\prime}$

where

$H=\frac{\partial\theta^{*}_{0}}{\partial\tilde{\theta}_{0}}=\begin{bmatrix}\frac{1}{\mathbb{E}[Y_{it}]}\cdot
I&\frac{-\theta_{0}}{[\mathbb{E}[Y_{it}]]^{2}}\end{bmatrix}$

and $\tilde{V}$ is as before, where the influence function $\psi_{0}$ is
vector-valued, corresponding to the vector $\theta_{0}$.

As an aside, when using OLS, the empirical influence function used in
estimating off-diagonal terms is

$\displaystyle\psi_{0}(W_{it})$
$\displaystyle=(\mathbb{E}[b_{it}b_{it}^{\prime-1}b_{it}\epsilon_{it}$
$\displaystyle\hat{\psi}_{it}$
$\displaystyle=\left(\frac{1}{\sum_{i=1}^{n}T_{i}}\sum_{i=1}^{n}\sum_{t\in
T_{i}}\sum_{s\in T_{i}}b_{it}b_{it}^{\prime}\right)^{-1}b_{it}\epsilon_{it}$

where $\epsilon_{it}$ is the OLS residual for observation $W_{it}$. As before,
we use a variance estimator that recognizes clustering.

##  Appendix E Additional Empirical Results

###  E.1 Regression Decomposition and ATT

spec. | treated | untreated | Lasso ATT | Lasso SE | RF ATT | RF SE | NN ATT | NN SE
---|---|---|---|---|---|---|---|---
1 | 185 | 254 | 1181.65 | 577.82 | 1627.93 | 584.52 | 1651.97 | 577.34
2 | 185 | 250 | 1191.35 | 580.46 | 1685.15 | 585.05 | 1603.91 | 582.80
3 | 185 | 179 | 1103.52 | 590.74 | 1935.65 | 575.73 | 1746.51 | 590.21
Table 7: ATT using NSW treatment and NSW control, by plug-in spec. | treated | untreated | Lasso ATT | Lasso SE | RF ATT | RF SE | NN ATT | NN SE
---|---|---|---|---|---|---|---|---
1 | 185 | 844 | 89.91 | 593.32 | 1214.55 | 617.46 | -1795.95 | 579.00
2 | 185 | 1620 | 492.77 | 594.27 | 926.93 | 624.56 | -0.35 | 595.95
3 | 185 | 2336 | 715.61 | 605.92 | 1001.66 | 609.32 | 1573.30 | 639.90
Table 8: ATT using NSW treatment and PSID comparison, by plug-in spec. | treated | untreated | Lasso ATT | Lasso SE | RF ATT | RF SE | NN ATT | NN SE
---|---|---|---|---|---|---|---|---
1 | 185 | 7687 | 195.91 | 595.90 | 1350.67 | 635.41 | 1345.07 | 605.18
2 | 185 | 13326 | 851.08 | 599.47 | 1434.67 | 621.56 | 1580.90 | 611.67
3 | 185 | 13449 | 979.20 | 605.02 | 1253.08 | 627.57 | 1350.68 | 668.13
Table 9: ATT using NSW treatment and CPS comparison, by plug-in

###  E.2 Panel Average Derivative and Demand Elasticities

We present elasticity estimates from OLS with a simpler specification than the
specification used in the main text. We take as $b_{1}(X_{it})$ the
concatenation of the following variables: log expenditure, and log price of
each good. For $\tilde{H}_{i}$, we use the time averages of $b_{1}(X_{it})$.
Note that $K=dim(b_{1}(X_{it}))=16$ and $p=dim(b_{it})=288$. We calculate
clustered standard errors derived by delta method as explained in Appendix D.
Tables 10 and 11 summarize results.

elasticity | point | SE
---|---|---
income | 0.42 | 0.05
own-price | -0.68 | 0.05
bread | -0.03 | 0.02
butter | 0.00 | 0.02
cereal | 0.00 | 0.02
chips | 0.02 | 0.03
coffee | 0.00 | 0.02
cookies | 0.00 | 0.02
eggs | -0.03 | 0.03
ice cream | -0.03 | 0.03
orange juice | -0.01 | 0.05
salad | 0.02 | 0.02
soda | -0.02 | 0.02
soup | -0.03 | 0.02
water | -0.01 | 0.02
yogurt | 0.01 | 0.04
Table 10: Milk elasticities, by OLS elasticity | point | SE
---|---|---
income | 0.64 | 0.02
own-price | -0.65 | 0.04
bread | -0.01 | 0.03
butter | -0.04 | 0.02
cereal | 0.01 | 0.03
chips | 0.02 | 0.03
coffee | 0.01 | 0.02
cookies | -0.03 | 0.02
eggs | -0.03 | 0.03
ice cream | 0.01 | 0.03
milk | 0.02 | 0.04
orange juice | -0.05 | 0.05
salad | -0.03 | 0.02
soup | 0.01 | 0.03
water | 0.01 | 0.02
yogurt | 0.05 | 0.04
Table 11: Soda elasticities from regression, by OLS

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