# Is $X(7200)$ the heavy anti-quark diquark symmetry partner of $X(3872)$?

Ming-Zhu Liu  School of Space and Environment, Beihang University, Beijing
100191, China  School of Physics, Beihang University, Beijing 100191, China
Li-Sheng Geng<EMAIL_ADDRESS>School of Physics, Beihang University,
Beijing 100191, China  Beijing Key Laboratory of Advanced Nuclear Materials
and Physics, Beihang University, Beijing 100191, China  Beijing Advanced
Innovation Center for Big Data-Based Precision Medicine, School of Medicine
and Engineering, Beihang University, Beijing, 100191  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

The $D^{(\ast)}\Xi_{cc}^{(\ast)}$ system and
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system can be related to the
$D^{(\ast)}\bar{D}^{(\ast)}$ system via heavy anti-quark di-quark symmetry
(HADS). In this work, we employ a contact-range effective field theory to
systematically investigate the likely existence of molecules in these systems
in terms of the hypothesis that X(3872) is a $1^{++}$ $D\bar{D}^{\ast}$ bound
state in the isospin symmetry limit, with some of the unknown low energy
constants estimated using the light-meson saturation approximation. In the
meson-meson system, a $J^{PC}=2^{++}$ $\bar{D}^{\ast}D^{\ast}$ molecule
commonly referred to as $X(4013)$ is reproduced, which is the heavy quark spin
partner of $X(3872)$. In the meson-baryon system, we predict two triply
charmed pentaquark molecules, $J^{P}=1/2^{-}$ $D^{\ast}\Xi_{cc}$ and
$J^{P}=5/2^{-}$ $D^{\ast}\Xi_{cc}^{\ast}$. In the baryon-baryon system, there
exist seven di-baryon molecules, $J^{PC}=0^{-+}$ $\bar{\Xi}_{cc}\Xi_{cc}$,
$J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}$, $J^{PC}=1^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=1^{--}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ and $J^{PC}=3^{--}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$. Among them, the $J^{PC}=0^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}$ and/or $J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}$
molecules may contribute to the $X(7200)$ state recently observed by the LHCb
Collaboration, which implies that $X(7200)$ can be related to $X(3872)$ via
HADS. As a byproduct, with the heavy quark flavor symmetry we also study
likely existence of molecular states in the $B^{(\ast)}\bar{B}^{(\ast)}$,
$\bar{B}^{(\ast)}\Xi_{bb}^{(\ast)}$, and
$\bar{\Xi}_{bb}^{(\ast)}\Xi_{bb}^{(\ast)}$ systems.

##  I Introduction

Hadrons can be classified either as mesons made of a pair of quark and
antiquark or as baryons made of three quarks in the conventional quark model,
which has been rather successful at least up to 2003. According to the quark
model Gell-Mann (1964), more complicated quark configurations, such as
multiquark states, should also exist, which are also allowed by QCD. In recent
years a lot of $XYZ$ and $P_{c}$ states that can not easily fit into the
conventional quark model were discovered by many collaborations all over the
world, which not only challenges the conventional quark model but also opens a
new era for hadron physics Chen  _et al._ (2016); Hosaka  _et al._ (2017); Guo
_et al._ (2018); Olsen  _et al._ (2018); Brambilla  _et al._ (2020).

In 2003, the Belle Collaboration reported a resonant state, $X(3872)$, in the
$J/\psi\pi^{+}\pi^{-}$ invariant mass spectrum of the $\Lambda_{b}\rightarrow
J/\psi\pi^{+}\pi^{-}K$ decay Aaij  _et al._ (2015), which has later been
confirmed by other collaborations Aubert  _et al._ (2005a); Acosta  _et al._
(2004); Abazov  _et al._ (2004); Aubert  _et al._ (2005b); Ablikim  _et al._
(2014); Aaij  _et al._ (2012). Its spin-parity was determined by the LHCb
Collaboration in 2013 Aaij  _et al._ (2013). Being close to the mass threshold
of $D\bar{D}^{\ast}$, it is naturally considered to be a hadronic molecule. In
addition, another convincing evidence of the molecular nature of $X(3872)$ is
its isospin breaking decays into $J/\Psi\pi\pi$ and $J/\Psi\pi\pi\pi$ Abe  _et
al._ (2005); del Amo Sanchez  _et al._ (2010). A series of theoretical works
interpreted $X(3872)$ as a $J^{PC}=1^{++}$ $\bar{D}D^{\ast}$ molecule Swanson
(2004); Voloshin (2004); AlFiky  _et al._ (2006); Liu  _et al._ (2008); Sun
_et al._ (2011); Nieves and Valderrama (2012); Guo  _et al._ (2013a); Karliner
and Rosner (2015); Liu  _et al._ (2019a). Although the molecular
interpretation seems to be the most popular, its nature has not been
completely unveiled. 111Other explanations also exist such as compact
tetraquark Maiani  _et al._ (2005); Ebert  _et al._ (2006); Matheus  _et al._
(2007); Wang and Huang (2014), charmonium Barnes and Godfrey (2004); Vijande
_et al._ (2005), molecule-charmonium Kalashnikova (2005); Matheus  _et al._
(2009); Ferretti  _et al._ (2013), and so on.  Nonetheless, given current
experimental and theoretical results, one may conclude that the molecule
components of $X(3872)$ must play a relevant (if not dominant) role. In our
previous work Liu  _et al._ (2019a), assuming that $X(3872)$ is a
$J^{PC}=1^{++}$ $\bar{D}D^{\ast}$ molecule, we have used the one boson
exchange model to predict the binding energies and scattering lengths of the
heavy quark spin symmetry (HQSS) partners of $X(3872)$.

HQSS plays an important role in describing the spectra of heavy hadrons
including both conventional and exotic ones. The $D$ and $D^{\ast}$ mesons as
well as $\Sigma_{c}$ and $\Sigma_{c}^{\ast}$ baryons can be regarded as
doublets of HQSS. Applying HQSS to their bottom partners, the mass splittings
of doublets become small, which indicates that HQSS works well in the heavy
quark limit. The interactions between heavy hadrons are also constrained by
HQSS, which can further decrease the number of unknown couplings and increase
the predictive power of effective field theories (EFT) as well. In the meson-
meson system, $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are assigned as $DK$ and
$D^{\ast}K$ hadronic molecules, respectively, belonging to a HQSS doublet
Barnes  _et al._ (2003); Kolomeitsev and Lutz (2004); Guo  _et al._ (2006);
Gamermann  _et al._ (2007); Faessler  _et al._ (2007a, b); Liu  _et al._
(2013); Altenbuchinger  _et al._ (2014). A lot of theoretical works described
$Z_{c}(3900)$ and $Z_{c}(4020)$ as isovector resonances generated by the
$\bar{D}D^{\ast}$ and $\bar{D}^{\ast}D^{\ast}$ interactions, respectively,
which indicates that $Z_{c}(3900)$ can be related to $Z_{c}(4020)$ via HQSS
Guo  _et al._ (2013a); Dong  _et al._ (2013); Wang  _et al._ (2020a). Very
recently $Z_{cs}(3985)$ was proposed to be the $SU(3)$-flavor partner of
$Z_{c}(3900)$ Yang  _et al._ (2020a); Meng  _et al._ (2020); Du  _et al._
(2020a). In the heavy meson and baryon systems, $P_{c}(4312)$, $P_{c}(4440)$,
and $P_{c}(4457)$ are nicely arranged into a multiplet of
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ hadronic molecules Liu  _et al._
(2019b); Du  _et al._ (2020b); Xiao  _et al._ (2019a); Sakai  _et al._ (2019);
Yamaguchi  _et al._ (2019); Lin and Zou (2019); Pavon Valderrama (2019); Liu
_et al._ (2019c), and in terms of HQSS there should exist other four states
which need to be confirmed by future experiments.

Heavy quark flavor symmetry has already shown its power in constructing the
spectra of charmed and bottom hadrons. In short, if there exists a charmed
hadron there should also exist the corresponding bottom partner. Heavy anti-
quark di-quark symmetry(HADS) dictates that in terms of color degree of
freedom a pair of heavy quarks can be related to a heavy antiquark in the
heavy quark limit Savage and Wise (1990). From this symmetry we can derive the
following mass relation:
$m_{\Xi_{cc}^{\ast}}-m_{\Xi_{cc}}=\frac{3}{4}(m_{D^{\ast}}-m_{D})$ Hu and
Mehen (2006). Although this relation has not been confirmed by experiments so
far, it is consistent with lattice QCD calculations and phenomenological
studies Padmanath  _et al._ (2015); Mathur and Padmanath (2019); Roncaglia
_et al._ (1995); Karliner and Rosner (2014). In our previous work, we have
employed HADS to extend the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ system to
the $\Xi_{cc}^{(\ast)}\Sigma_{c}^{(\ast)}$ system, and predicted the likely
existence of $\Xi_{cc}^{(\ast)}\Sigma_{c}^{(\ast)}$ bound states according to
the hypothesis that $P_{c}(4440)$ and $P_{c}(4457)$ are
$\bar{D}^{\ast}\Sigma_{c}$ bound states Pan  _et al._ (2019, 2020).

Figure 1: HADS related hidden charm tetraquark states, triply charmed
pentaquark states, and hidden charmed hexaquark states.

Along the line of Ref. Pan  _et al._ (2019), we can extend the
$D^{(\ast)}\bar{D}^{(\ast)}$ system to the $D^{(\ast)}\Xi_{cc}^{(\ast)}$
system and then to the $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system via
HADS as shown in Fig. 1, and systematically investigate the likely existence
of bound states in these systems using a contact-range EFT. In Refs. Guo  _et
al._ (2013a, b) the authors have already studied the
$D^{(\ast)}\bar{D}^{(\ast)}$ and $D^{(\ast)}\Xi_{cc}^{(\ast)}$ systems in the
contact-range EFT. However, because of the existence of two unknown couplings
and only one effective input, the full spectra of these two systems could not
be determined. In this work we employ the light-meson saturation approach to
estimate the ratio of the two couplings Peng  _et al._ (2020a). In our
previous work Liu  _et al._ (2019c), we found that the two couplings of the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ system determined by the contact-range
EFT and the light-meson saturation approach are quite similar. In Ref. Peng
_et al._ (2020b) Peng. et al used the light-meson saturation approach to
estimate the contact-range potential between $\bar{D}\Xi_{c}$ and
$\bar{D}\Xi_{c}^{\prime}$, yielding results consistent with the local hidden
gauge approach Xiao  _et al._ (2019b); Liu  _et al._ (2020). Therefore, in
this work, to fully determine the mass spectra of these systems, we will
combine the hypothesis that $X(3872)$ is a $J^{PC}=1^{++}$ bound state and the
light-meson saturation approach to determine the values of the only two
unknown couplings $C_{a}$ and $C_{b}$, then calculate the binding energies (if
bound states exist) and scattering lengths of the
$D^{(\ast)}\bar{D}^{(\ast)}$, $D^{(\ast)}\Xi_{cc}^{(\ast)}$, and
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ systems.

Recently, the LHCb Collaboration reported one resonant state with a mass of
6.9 GeV in the $J/\psi$-$J/\psi$ invariant mass spectrum, labelled as
$X(6900)$. In addition, there exist two more structures, a broad one from 6.2
to 6.8 GeV and a vague structure around 7.2 GeV Aaij  _et al._ (2020). These
tetraquark states containing fully charmed quarks undoubtedly open another new
era for hadron physics since the discovery of $X(3872)$, which has motivated a
lot of theoretical studies Ma and Zhang (2020); Dong  _et al._ (2020); Wang
(2020); Karliner and Rosner (2020); Maciuła  _et al._ (2020); Chao and Zhu
(2020); Richard (2020); Maiani (2020); Sonnenschein and Weissman (2020); Giron
and Lebed (2020); Wang  _et al._ (2020b); Weng  _et al._ (2020); Zhu (2020);
Guo and Oller (2020); Zhu  _et al._ (2020); Cao  _et al._ (2020); liu  _et
al._ (2020); Gong  _et al._ (2020). Two popular interpretation of these
structures are resonances generated by several coupled channels of charmonium
pairs and compact tetraquark states of diquark and anti-diquark. In this work
we mainly focus on the structure near 7.2 GeV, namely $X(7200)$, whose mass is
close to the mass threshold of $\bar{\Xi}_{cc}\Xi_{cc}$. Since the
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system can be related to the
$\bar{D}^{(\ast)}D^{(\ast)}$ system via HADS, we further explore whether there
exist $\bar{\Xi}_{cc}\Xi_{cc}$ molecules from the hypothesis that $X(3872)$ is
a $J^{PC}=1^{++}$ bound state, in other words, whether $X(7200)$ can be
related to $X(3872)$ via HADS.

The manuscript is structured as follows. In Sec. II we present the contact-
range potentials of the $D^{(\ast)}\bar{D}^{(\ast)}$,
$D^{(\ast)}\Xi_{cc}^{(\ast)}$, and $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$
systems. In Sec. III we introduce the light-meson saturation approach to help
determine the unknown couplings. Then we calculate the binding energies (if
bound states exist) and scattering lengths of the
$D^{(\ast)}\bar{D}^{(\ast)}$, $D^{(\ast)}\Xi_{cc}^{(\ast)}$ and
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ systems. Finally we present the
conclusions in Sec. IV

##  II Formalism

Generically, the interaction between two heavy hadrons can be decomposed into
a long- and a short-range piece within the EFT. The long-range piece
corresponds to the one-pion-exchange potential, while the short-range piece
can be described by a series of contact-range potentials with unknown
couplings. In the present context, the short-range interaction between two
heavy hadrons is constrained by heavy quark symmetry. In the following we
explain how to derive the $D^{(\ast)}\bar{D}^{(\ast)}$,
$D^{(\ast)}\Xi_{cc}^{(\ast)}$, and $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$
interactions. In the line of Refs. Liu  _et al._ (2020); Pan  _et al._ (2019),
their interactions can be determined in the EFT approach. One should note that
we just consider the leading order contact-range potentials because our
previous studies indicated that the pion exchanges are perturbative in the
charm sector Valderrama (2012); Lu  _et al._ (2019). The short-range
interaction of the meson-meson system can be written as

$\displaystyle\mathcal{L}_{HH}=C_{a}Tr[H^{{\dagger}}H]Tr[H^{\prime{\dagger}}H^{\prime}]+C_{b}Tr[H^{{\dagger}}\sigma_{i}H]Tr[H^{\prime{\dagger}}\sigma_{i}H^{\prime}],$
(1)

where $C_{a}$ and $C_{b}$ are two unknown couplings and $H$ is a non-
relativistic superfield with
$H=\frac{1}{\sqrt{2}}[P+\vec{P}^{\ast}\cdot\vec{\sigma}]$, and $P$ and
$P^{\ast}$ denote pseuoscalar and vector charmed meson fields. The
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ system can be related to the
$D^{(\ast)}\bar{D}^{(\ast)}$ system through HADS, resulting in the following
Lagrangian

$\displaystyle\mathcal{L}_{HT}=C_{a}Tr[H^{{\dagger}}H]Tr[T^{\prime{\dagger}}T^{\prime}]+C_{b}Tr[H^{{\dagger}}\sigma_{i}H]Tr[T^{\prime{\dagger}}\sigma_{i}T^{\prime}],$
(2)

where $T=\frac{1}{\sqrt{3}}\vec{\sigma}\Xi_{cc}+\vec{\Xi}_{cc}^{\ast}$ is a
non-relativistic superfield of $\Xi_{cc}$ and $\Xi_{cc}^{\ast}$ respecting
HQSS. The Lagrangian of the $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system
have the same form as those of the $D^{(\ast)}\bar{D}^{(\ast)}$ and
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ systems

$\displaystyle\mathcal{L}_{TT}=C_{a}Tr[T^{{\dagger}}T]Tr[T^{\prime{\dagger}}T^{\prime}]+C_{b}Tr[T^{{\dagger}}\sigma_{i}T]Tr[T^{\prime{\dagger}}\sigma_{i}T^{\prime}].$
(3)

The term $C_{a}$ is independent of the spin, while $C_{b}$ accounts for the
spin spin interaction. Using the same approach of Ref. Pan  _et al._ (2019),
the contact potentials of the $D^{(\ast)}\bar{D}^{(\ast)}$,
$D^{(\ast)}\Xi_{cc}^{(\ast)}$, and $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$
systems can be easily constructed and are shown in Table 1.

Table 1: Contact-range potentials for the heavy meson-heavy anti-meson, heavy meson-doubly heavy baryon, and doubly heavy baryon-doubly heavy anti-baryon systems depending on two unknown coupling constants, $C_{a}$ and $C_{b}$. These coupling constants can be determined from the sum of $C_{a}$ and $C_{b}$ by reproducing the mass of $X(3872)$ and the ratio of $C_{a}$ and $C_{b}$ by the light-meson saturation approach.  state | $J^{PC}$ | V | state | $J^{P}$ | V | state | $J^{PC}$ | V
---|---|---|---|---|---|---|---|---
$D\bar{D}$ | $0^{++}$ | $C_{a}$ | $D\Xi_{cc}$ | $\frac{1}{2}^{-}$ | $C_{a}$ | $\bar{\Xi}_{cc}\Xi_{cc}$ | $0^{-+}$ | $C_{a}-\frac{1}{3}C_{b}$
$\bar{\Xi}_{cc}\Xi_{cc}$ | $1^{--}$ | $C_{a}+\frac{1}{9}C_{b}$
$D^{\ast}\bar{D}$/$D\bar{D}^{\ast}$ |  |  |  |  |  | $\bar{\Xi}_{cc}^{\ast}\Xi_{cc}/\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$ | $1^{-+}$ | $C_{a}+C_{b}$
$1^{++}$ | $C_{a}+C_{b}$ | $D^{\ast}\Xi_{cc}$ | $\frac{1}{2}^{-}$ | $C_{a}+\frac{2}{3}C_{b}$ | $1^{--}$ | $C_{a}+\frac{1}{9}C_{b}$
$1^{+-}$ | $C_{a}-C_{b}$ | $\frac{3}{2}^{-}$ | $C_{a}-\frac{1}{3}C_{b}$ | $2^{-+}$ | $C_{a}+C_{b}$
|  |  |  |  | $2^{--}$ | $C_{a}-\frac{5}{3}C_{b}$
|  |  | $D\Xi_{cc}^{\ast}$ | $\frac{3}{2}^{-}$ | $C_{a}$ | $\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ | $0^{-+}$ | $C_{a}-\frac{5}{3}C_{b}$
$D^{\ast}\bar{D}^{\ast}$ | $0^{++}$ | $C_{a}-2C_{b}$ | $D^{\ast}\Xi_{cc}^{\ast}$ | $\frac{1}{2}^{-}$ | $C_{a}-\frac{5}{3}C_{b}$ | $1^{--}$ | $C_{a}-\frac{11}{9}C_{b}$
$1^{+-}$ | $C_{a}-C_{b}$ | $\frac{3}{2}^{-}$ | $C_{a}-\frac{2}{3}C_{b}$ | $2^{-+}$ | $C_{a}-\frac{1}{3}C_{b}$
$2^{++}$ | $C_{a}+C_{b}$ | $\frac{5}{2}^{-}$ | $C_{a}+C_{b}$ | $3^{--}$ | $C_{a}+C_{b}$

To make concrete predictions for likely existence of molecular states we have
to solve a non-relativistic bound state equation with the contact-range
potentials of Table 1. In the momentum space, we can solve the Lippmann-
Schinwinger equation to find bound state as

$\displaystyle\phi(k)+\int\frac{d^{3}p}{(2\pi)^{3}}\langle
k|V|p\rangle\frac{\phi(p)}{B+\frac{p^{2}}{2\mu}}=0,$ (4)

where $\phi(k)$ is the vertex function, $B$ the binding energy, and $\mu$ the
reduced mass. To solve the equation we regularize the contact potential in the
following way

$\displaystyle\langle
p|V_{\Lambda}|p^{\prime}\rangle=C(\Lambda)f(\frac{p}{\Lambda})f(\frac{p^{\prime}}{\Lambda}),$
(5)

with $\Lambda$ the cutoff, $f(x)$ a regular function, and $C({\Lambda})$ the
running coupling constant. A typical choice of the cutoff is $\Lambda=0.5-1$
GeV, while for the regulator we choose a Gaussian type $f(x)=e^{-x^{2}}$. In
this work, to be consistent with the cutoff adopted in the light-meson
saturation approach, we use a cutoff of $\Lambda=0.7$ GeV. Considering only
$S$-wave contact interactions, the integral equation simplifies to

$\displaystyle
1+C(\Lambda)\frac{\mu}{\pi^{2}}\int_{0}^{\infty}dqe^{-2\frac{q^{2}}{\Lambda^{2}}}\frac{q^{2}}{B+\frac{\vec{q}^{2}}{2\mu}}=0.$
(6)

##  III Numerical results and discussions

In this section we study the likely existence of molecular states in the
$D^{(\ast)}\bar{D}^{(\ast)}$, $D^{(\ast)}\Xi_{cc}^{(\ast)}$, and
$\Xi_{cc}^{(\ast)}\bar{\Xi}_{cc}^{(\ast)}$ systems. The contact-range
potentiasl of these systems are only dependent on two couplings, $C_{a}$ and
$C_{b}$. To determine these couplings, the best approach is to fit to the
experimental data. However, there exists only one molecular candidate
$X(3872)$ to determine the sum of $C_{a}$ and $C_{b}$. We, therefore, resort
to the light-meson saturation approach to estimate the ratio of $C_{a}$ and
$C_{b}$.

The assumption of light-meson saturation implies that the values of the EFT
couplings are saturated by light meson exchanges. Following the formalism of
Ref. Peng  _et al._ (2020a), we estimate the couplings of $C_{a}$ and $C_{b}$
by the sigma and vector meson exchange saturation,

$\displaystyle C_{a}^{sat}(\Lambda\sim m_{\sigma},m_{V})$
$\displaystyle\propto$ $\displaystyle C_{a}^{S}+C_{a}^{V},$ (7) $\displaystyle
C_{b}^{sat}(\Lambda\sim m_{\sigma},m_{V})$ $\displaystyle\propto$
$\displaystyle C_{b}^{V},$

where the cutoff $\Lambda\sim m_{\sigma},m_{V}$ implies that the saturation
works at an EFT cutoff close to the masses of exchanged mesons, i.e., $0.6\sim
0.8$ GeV. The values of saturated couplings are expected to be proportional to
the potentials of light meson exchanges in the OBE model once we have removed
the spurious Dirac-delta potential. Thus $C_{a}$ and $C_{b}$ can be written as

$\displaystyle C_{a}^{sat(\sigma)}(\Lambda\sim m_{\sigma})$
$\displaystyle\propto$ $\displaystyle-\frac{g_{\sigma}^{2}}{m_{\sigma}^{2}},$
(8) $\displaystyle C_{a}^{sat(V)}(\Lambda\sim m_{V})$ $\displaystyle\propto$
$\displaystyle-\frac{g_{v}^{2}}{m_{v}^{2}}(1+\vec{\tau}_{1}\cdot\vec{\tau}_{2}),$
$\displaystyle C_{b}^{sat(V)}(\Lambda\sim m_{V})$ $\displaystyle\propto$
$\displaystyle-\frac{f_{v}^{2}}{4M^{2}}(1+\vec{\tau}_{1}\cdot\vec{\tau}_{2}),$

where $g_{\sigma_{1}}$ denotes the charm meson coupling to sigma meson,
$g_{v1}$ and $f_{v1}$ denote electric-type and magnetic-type couplings between
charm mesons and light vector mesons, and $M$ is a mass scale to render
$f_{v1}$ dimensionless, see, e.g., Table 2. The proportionality constant is
unknown and depends on the details of the renormalization procedure. However,
assuming that the constant is the same for $C_{a}^{sat}$ and $C_{b}^{sat}$, we
can calculate their ratio.

Table 2: Masses of light mesons, heavy mesons, and heavy baryons as well as couplings of the heavy mesons to the light mesons. The magnetic couplings of vector mesons are defined as $f_{v}=\kappa_{v}g_{v}$, and $M$ refers to the mass scale (in MeV) involved in the magnetic-type couplings.  Coupling | Value for $D^{(\ast)}$ | Light meson | M(MeV) | Heavy meson | M(MeV) | Heavy baryon | M(MeV)
---|---|---|---|---|---|---|---
$g_{\sigma}$ | 3.4 | $\pi$ | 138 | $D$ | 1867 | $\Xi_{cc}$ | 3621
$g_{v}$ | 2.6 | $\sigma$ | 600 | $D^{\ast}$ | 2009 | $\Xi_{cc}^{\ast}$ | 3727Hu and Mehen (2006)
$\kappa_{v}$ | 2.4 | $\rho$ | 770 | $B$ | 5279 | $\Xi_{bb}$ | 10127Lewis and Woloshyn (2009)
$M$ | 940 | $\omega$ | 780 | $B^{\ast}$ | 5325 | $\Xi_{bb}^{\ast}$ | 10151Lewis and Woloshyn (2009)

In the following, we explain how to determine the values of $C_{a}$ and
$C_{b}$ in detail. Assuming that $X(3872)$ is a $J^{PC}=1^{++}$
$D^{\ast}\bar{D}$ bound state with the contact potential $C_{a}+C_{b}$, we can
determine the sum of $C_{a}$ and $C_{b}$ as 31.4 GeV-2 by Eq. (6), where the
cutoff is fixed at 0.7 GeV. With this sum we can only predict the binding
energies of states for which the contact potentials are $C_{a}+C_{b}$.
Fortunately, combining the ratio of $C_{a}$ and $C_{b}$ determined by the
light-meson saturation, we can determine the values of $C_{a}$ and $C_{b}$.
The ratio is

$\displaystyle\frac{C_{b}^{sat}}{C_{a}^{sat}}=\frac{C_{b}^{sat(V)}}{C_{a}^{sat(V+\sigma)}}\approx
0.35.$ (9)

Comparing the ratio $C_{b}/C_{a}\approx 0.134$ in the meson-baryon system of
Ref. Liu  _et al._ (2019c), we find that the spin-spin term plays a more
important role in the meson-meson system, which could be the main reason why
we can not obtain a complete multiplet of hadronic molecules in the heavy
meson-meson system. Using the ratio determined by the light-meson saturation
we finally obtain $C_{a}=-23.3$ GeV-2 and $C_{b}=-8.1$ GeV-2, which can help
us compute the full spectra of the $D^{(\ast)}\bar{D}^{(\ast)}$,
$D^{(\ast)}\Xi_{cc}^{(\ast)}$, and $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$
systems in the EFT approach. According to the EFT, the breaking of HADS is of
the order of $\Lambda_{QCD}/(m_{Q}v)$ Savage and Wise (1990), where
$\Lambda_{QCD}\sim 200-300$ MeV and $m_{Q}$ and $v$ are the mass and velocity
of the di-quark. From the estimation of Ref. Cho (1993), $m_{Q}v\sim$0.8 GeV
for a charm quark pair. In this work we take $30\%$ uncertainty for HADS. The
breaking of HQSS is taken to be $15\%$ Pan  _et al._ (2019).

###  III.1  Meson-meson system

Table 3:  Scattering lengths ($a$ in fm), binding energies ($B$ in MeV if bound states exist) and mass spectra ($M$ in MeV) of prospective isoscalar heavy antimeson-meson molecules. The $c$ and $b$ subscripts denote the charm and bottom sector, respectively. The uncertainties originate from the HQSS breaking of the order $15\%$.  molecule | $I$ | $J^{PC}$ |  $a_{c}$ (fm) |  $B_{c}$ (MeV) |  $M_{c}$ (MeV ) | molecule |  $a_{b}$ (fm) |  $B_{b}$ (MeV) |  $M_{b}$ (MeV )
---|---|---|---|---|---|---|---|---|---
$D\bar{D}$ | $0$ | $0^{++}$ | $-20.5_{+27.5}^{+17.2}$ | ${\dagger}$ | ${\dagger}$ | $B\bar{B}$ | $1.1_{-0.1}^{+0.1}$ | $20.1^{+7.3}_{-6.7}$ | $10537.9$
$D\bar{D}^{\ast}+D^{\ast}\bar{D}$ | $0$ | $1^{++}$ | $2.7_{-0.7}^{+2.8}$ | Input | Input | $B\bar{B}^{\ast}+B^{\ast}\bar{B}$ | $1.0_{+0.0}^{-0.0}$ | $37.9^{+10.9}_{-10.4}$ | 10566.1
$D\bar{D}^{\ast}-D^{\ast}\bar{D}$ | $0$ | $1^{+-}$ | $-1.3_{-0.8}^{+0.4}$ | ${\dagger}$ | ${\dagger}$ | $B\bar{B}^{\ast}-B^{\ast}\bar{B}$ | $1.6_{-0.2}^{+0.5}$ | $5.7^{+3.5}_{-2.9}$ | $10598.3$
$D^{\ast}\bar{D}^{*}$ | $0$ | $0^{++}$ | $-0.3^{-0.1}_{+0.1}$ | ${\dagger}$ | ${\dagger}$ | $B^{\ast}\bar{B}^{*}$ | $-3.1^{+1.5}_{-6.9}$ | ${\dagger}$ | ${\dagger}$
$D^{\ast}\bar{D}^{*}$ | $0$ | $1^{+-}$ | $-1.5_{-1.0}^{+0.5}$ | ${\dagger}$ | ${\dagger}$ | $B^{\ast}\bar{B}^{*}$ | $1.6^{+0.5}_{-0.2}$ | $5.8_{-3.0}^{+3.5}$ | $10644.2$
$D^{\ast}\bar{D}^{*}$ | $0$ | $2^{++}$ | $2.5_{-0.6}^{+1.9}$ | $4.9^{+5.3}_{-3.6}$ | $4013.1$ | $B^{\ast}\bar{B}^{*}$ | $1.0_{-0.1}^{+0.0}$ | $38.1^{+10.9}_{-10.5}$ | $10611.9$

The binding energies (if bound states exist) and scattering lengths of the
$D^{(\ast)}\bar{D}^{(\ast)}$ system are shown in Table 3. For the
$D^{(\ast)}\bar{D}^{(\ast)}$ system, we obtain a $J^{PC}=2^{++}$
$D^{\ast}\bar{D}^{\ast}$ bound state with a binding energy of 5 MeV, namely
$X(4013)$, which has already been predicted by many theoretical studies Nieves
and Valderrama (2012); Guo  _et al._ (2013a); Baru  _et al._ (2016). However,
its existence has not been confirmed by experiments. Except for $X(3872)$ and
$X(4013)$ there are no other bound states in this system, which is consistent
with the conclusion of the OBE model Liu  _et al._ (2019a). From the
scattering length of $J^{PC}=0^{++}$ $\bar{D}D$ we find that the potential
between $\bar{D}$ and $D$ is attractive, but is too weak to form a bound
state. Recently a Lattice QCD study obtained a shallow bound state located 4
MeV below the $D\bar{D}$ mass threshold Prelovsek  _et al._ (2020). If we use
the OBE model to reproduce the 4 MeV binding energy, the cutoff needed is
about $\Lambda=1.4$ GeV, whose value is close to but larger than that obtained
in our previous work to reproduce $X(3872)$, i.e., $\Lambda=1.01$ GeV. We note
that several theoretical works have predicted the existence of a bound state
in the $D\bar{D}$ channel Gamermann  _et al._ (2007); Sun  _et al._ (2011);
Wang  _et al._ (2020c); Ding  _et al._ (2020); Dai  _et al._ (2020).

With heavy quark flavor symmetry we can also study the
$B^{(\ast)}\bar{B}^{(\ast)}$ system. We find 5 bound states, $J^{PC}=0^{++}$
$\bar{B}B$, $J^{PC}=1^{++}$ $\bar{B}B^{\ast}$, $J^{PC}=1^{+-}$
$\bar{B}B^{\ast}$, $J^{PC}=1^{+-}$ $\bar{B}^{\ast}B^{\ast}$, and
$J^{PC}=2^{++}$ $\bar{B}^{\ast}B^{\ast}$, consistent with the results of the
OBE model Liu  _et al._ (2019a). Among them we can regard the $J^{PC}=1^{++}$
$\bar{B}B^{\ast}$ and $J^{PC}=2^{++}$ $\bar{B}^{\ast}B^{\ast}$ states,
referred to as $X_{b}$ and $X_{b}^{\prime}$, as the hidden bottom partners of
$X(3872)$ and $X(4013)$, respectively. We note that Törnqbist has already
predicted $X(4013)$ with a binding energy 45 MeV in the one pion exchange
model Tornqvist (1994), but its existence has not been confirmed by
experiments.

###  III.2 Meson-baryon system

Table 4:  Scattering lengths ($a$ in fm), binding energies ($B$ in MeV, if bound states exist) and mass spectra ($M$ in MeV) of prospective isoscalar heavy meson-baryon molecules. The $c$ and $b$ subscript denote the charm and bottom sector, respectively. The uncertainties originate from the HADS breaking of the order $30\%$.  molecule | $I$ | $J^{PC}$ |  $a_{c}$ (fm) |  $B_{c}$ (MeV) |  $M_{c}$ (MeV ) | molecule |  $a_{b}$ (fm) |  $B_{b}$ (MeV) |  $M_{b}$ (MeV )
---|---|---|---|---|---|---|---|---|---
$D\Xi_{cc}$ | $0$ | $1/2^{-}$ | $3.3^{-9.1}_{-1.5}$ | $2.0^{+7.3}_{{\dagger}}$ | ? | $\bar{B}\Xi_{bb}$ | $1.0_{-0.1}^{+0.2}$ | $27.2^{+16.3}_{-14.7}$ | 15378.8
$D\Xi_{cc}^{\ast}$ | $0$ | $3/2^{-}$ | $3.2^{-9.5}_{-1.4}$ | $2.1^{+7.5}_{{\dagger}}$ | ? | $\bar{B}\Xi_{bb}^{\ast}$ | $1.0_{-0.1}^{+0.2}$ | $27.2^{+16.3}_{-14.7}$ | 15402.8
$D^{\ast}\Xi_{cc}$ | $0$ | $1/2^{-}$ | $1.8_{-0.5}^{+3.5}$ | $8.7^{+13.3}_{-8.1}$ | 5621.3 | $\bar{B}^{\ast}\Xi_{bb}$ | $0.9_{-0.0}^{+0.1}$ | $40.0^{+21.0}_{-19.6}$ | 15412.0
$D^{\ast}\Xi_{cc}$ | $0$ | $3/2^{-}$ | $4.7^{-8.0}_{-2.7}$ | $0.9^{+5.3}_{{\dagger}}$ | $?$ | $\bar{B}^{\ast}\Xi_{bb}$ | $1.0^{-0.1}_{+0.3}$ | $21.3^{+13.9}_{-12.2}$ | $15430.7$
$D^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $1/2^{-}$ | $-0.9_{-0.9}^{+0.4}$ | ${\dagger}$ | ${\dagger}$ | $\bar{B}^{\ast}\Xi_{bb}^{\ast}$ | $2.1^{+15.2}_{-0.6}$ | $1.9^{+4.0}_{-1.9}$ | $15474.0$
$D^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $3/2^{-}$ | $22.8_{-22.0}^{-24.6}$ | $0^{+2.8}_{{\dagger}}$ | $?$ | $\bar{B}^{\ast}\Xi_{bb}^{\ast}$ | $1.1_{-0.1}^{+0.4}$ | $15.7^{+11.4}_{-10.8}$ | $15460.3$
$D^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $5/2^{-}$ | $1.6_{-0.4}^{+1.7}$ | $12.9^{+15.9}_{-10.9}$ | $5723.2$ | $\bar{B}^{\ast}\Xi_{bb}^{\ast}$ | $0.9_{-0.1}^{+0.1}$ | $46.6^{+23.3}_{-22.0}$ | $15429.4$

Using HADS we can extend the $D^{(\ast)}\bar{D}^{(\ast)}$ system to the
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ system as shown in Fig. 1, where a HQSS
multiplet of meson-baryon molecules can be expected. The contact-range
potentials of the $D^{(\ast)}\Xi_{cc}^{(\ast)}$ system are expressed as
combinations of $C_{a}$ and $C_{b}$, which can be related to the couplings
arising in the $D^{(\ast)}\bar{D}^{(\ast)}$ system by HADS. Therefore, we take
the same input to calculate the binding energies and scattering lengths of the
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ system, and the results are given in Table 4.
Although most $D^{(\ast)}\Xi_{cc}^{(\ast)}$ systems are bound with the central
values of $C_{a}$ and $C_{b}$, they may become unbound once the breaking of
HADS is taken into account. Only two systems, $J^{P}=1/2^{-}$
$D^{\ast}\Xi_{cc}$ and $J^{P}=5/2^{-}$ $D^{\ast}\Xi_{cc}^{\ast}$, always
remain bound. Thus in this work we predict the existence of two triply charmed
pentaquark molecules. It is to be noted that in our previous study, we also
predicted a $J^{P}=1/2^{-}$ $D^{\ast}\Xi_{cc}$ bound state in the OBE model
Liu  _et al._ (2018), while in Ref. Guo  _et al._ (2013b) a $J^{P}=5/2^{-}$
$D^{\ast}\Xi_{cc}^{\ast}$ bound state is predicted using a contact-range
potential. These two molecules can decay into $\Omega_{ccc}\omega$ via
triangle diagrams, and we expect that they can be detected in future
experiments.

With heavy quark flavor symmetry we also study the
$\bar{B}^{(\ast)}\Xi_{bb}^{(\ast)}$ system, and find that bound states exist
in all the channels within uncertainties induced by heavy quark symmetry
breaking, which indicates that we obtain a complete HQSS multiplet of hadronic
molecules. Comparing the results with those of the meson-meson system, we find
that the meson-baryon molecular states are more bound, in agreement with naive
expectations because of the smaller reduced masses in the latter system.

Table 5:  Scattering lengths ($a$ in fm), binding energies ($B$ in MeV, if bound states exist) and mass spectra ($M$ in MeV) of prospective isoscalar heavy baryon-baryon molecules. The $c$ and $b$ subscript denote the charm and bottom sector, respectively. The uncertainties originate from the HADS breaking of the order $30\%$.  molecule | $I$ | $J^{PC}$ |  $a_{c}$ (fm) |  $B_{c}$ (MeV) |  $M_{c}$ (MeV ) | molecule |  $a_{b}$ (fm) |  $B_{b}$ (MeV) |  $M_{b}$ (MeV )
---|---|---|---|---|---|---|---|---|---
$\bar{\Xi}_{cc}\Xi_{cc}$ | $0$ | $0^{-+}$ | $1.8_{-0.5}^{+3.4}$ | $6.3^{+9.4}_{-5.8}$ | 7235.7 | $\bar{\Xi}_{bb}\Xi_{bb}$ | $0.9_{-0.1}^{+0.1}$ | $29.3^{+15.1}_{-14.1}$ | 20224.7
$\bar{\Xi}_{cc}\Xi_{cc}$ | $0$ | $1^{--}$ | $1.5^{+1.2}_{-0.3}$ | $11.5^{+12.9}_{-9.3}$ | 7230.4 | $\bar{\Xi}_{bb}\Xi_{bb}$ | $0.9^{+0.1}_{-0.1}$ | $38.2^{+18.2}_{-17.3}$ | 20215.8
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}$+$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$ | $0$ | $1^{-+}$ | $1.2^{+0.4}_{-0.2}$ | $25.0^{+19.7}_{-16.4}$ | 7323.0 | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}+\bar{\Xi}_{bb}\Xi_{bb}^{\ast}$ | $0.8^{+0.1}_{-0.0}$ | $56.5^{+24.4}_{-23.6}$ | 20221.5
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}-\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$ | $0$ | $1^{--}$ | $1.4^{+1.2}_{-0.2}$ | $11.9^{+13.1}_{-9.5}$ | 7336.1 | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}-\bar{\Xi}_{bb}\Xi_{bb}^{\ast}$ | $0.9^{+0.1}_{-0.1}$ | $38.2^{+18.2}_{-17.3}$ | 20245.9
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}+\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$ | $0$ | $2^{-+}$ | $1.2^{+0.4}_{-0.2}$ | $25.0^{+19.7}_{-16.4}$ | 7323.0 | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}+\bar{\Xi}_{bb}\Xi_{bb}^{\ast}$ | $0.8^{+0.1}_{-0.0}$ | $56.5^{+24.4}_{-23.6}$ | 20221.5
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}-\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$ | $0$ | $2^{--}$ | $-2.6^{+42.6}_{+1.7}$ | ${\dagger}$ | ${\dagger}$ | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}-\bar{\Xi}_{bb}\Xi_{bb}^{\ast}$ | $1.3^{+0.8}_{-0.2}$ | $5.7_{-4.2}^{+5.5}$ | 20272.3
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $0^{-+}$ | $-2.7^{+24.8}_{+1.8}$ | ${\dagger}$ | ${\dagger}$ | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}^{\ast}$ | $1.3^{+0.8}_{-0.2}$ | $5.7_{-4.2}^{+5.5}$ | 20296.3
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $1^{--}$ | $8.0^{-10.3}_{-5.6}$ | $0.2^{+2.8}_{{\dagger}}$ | $?$ | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}^{\ast}$ | $1.1^{+0.3}_{-0.1}$ | $12.8^{+8.8}_{-7.6}$ | 20289.2
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $2^{-+}$ | $1.7_{-0.4}^{+2.7}$ | $6.9^{+9.7}_{-6.2}$ | $7447.1$ | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}^{\ast}$ | $0.9^{+0.1}_{-0.1}$ | $29.4^{+15.0}_{-14.2}$ | 20272.6
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ | $0$ | $3^{--}$ | $1.1^{+0.5}_{-0.1}$ | $25.6^{+19.7}_{-16.6}$ | $7428.4$ | $\bar{\Xi}_{bb}^{\ast}\Xi_{bb}^{\ast}$ | $0.8^{+0.1}_{-0.0}$ | $56.5^{+24.4}_{-23.5}$ | 20245.5

###  III.3 Baryon-anti-baryon system

In this section, we further extend the $D^{(\ast)}\Xi_{cc}^{(\ast)}$ system to
the $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system as shown in Fig. 1. This
baryon-antibaryon system contains 10 channels related by HQSS. The contact-
range potentials of the $\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system can
also be written in terms of $C_{a}$ and $C_{b}$ dictated by HQSS. We compute
the binding energies and scattering lengths of the
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system, and show the results in
Table 5. We find seven bound states, i.e., $J^{PC}=0^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}$, $J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}$,
$J^{PC}=1^{-+}$ $\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=1^{--}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$, and $J^{PC}=3^{--}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$. The $\bar{\Xi}_{cc}\Xi_{cc}$ system
Meng  _et al._ (2017) and the $\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ system
Yang  _et al._ (2020b) have been studied in the OBE model, which show that all
of these systems are bound. Assuming heavy quark flavor symmetry, we further
investigate their hidden bottom partners
$\bar{\Xi}_{bb}^{(\ast)}\Xi_{bb}^{(\ast)}$, and all the states are found to be
bound as shown in Table 5, suggesting the emergence of a complete multiplet of
hadronic molecules.

Figure 2: Left: the two charmed meson channel heavily affects the mass spectra
of charmonia close to their mass thresholds. Right: HADS implies that the two
doubly charmed baryon channel could also affect the mass spectra of fully
charmed tetraquark states close to their mass thresholds.

Among these molecules, we note that there are two molecules, $J^{PC}=0^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}$ and $J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}$ ,
located close to 7200 MeV, which could contribute to the structure $X(7200)$
observed by the LHCb Collaboration recently. These two molecules are predicted
in terms of the hypothesis that $X(3872)$ is a $J^{PC}=1^{++}$
$\bar{D}D^{\ast}$ bound state and utilyzing the HADS. It is well known that
the heavy meson-anti meson channel affects the spectra of conventional
charmonia. According to the relationship between $X(7200)$ and $X(3872)$
proposed by the present work it is rather likely that the two doubly charmed
baryon-antibaryon channel would affect the spectra of fully charmed tetraquark
states as shown in Fig. 2, which indicates that $X(7200)$ might contain both
$\bar{\Xi}_{cc}\Xi_{cc}$ molecular and charmed diquark-anti diquark tetraquark
components. It is interesting to note that our study shows that $X(7200)$
might be a superposition of two molecular states.

##  IV Summary

In this work we extended the $D^{(\ast)}\bar{D}^{(\ast)}$ system to the
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ system then to the
$\bar{\Xi}_{cc}^{(\ast)}\Xi_{cc}^{(\ast)}$ system utilyzing the heavy
antiquark diquark symmetry (HADS). The contact-range potentials of these
systems respecting heavy quark spin symmetry (HQSS) are constrained to two
couplings at the leading order, $C_{a}$ and $C_{b}$. Assuming that $X(3872)$
is a $J^{PC}=1^{++}$ $\bar{D}D^{\ast}$ bound state, we determined the sum of
$C_{a}$ and $C_{b}$. To fully determine the value of $C_{a}$ and $C_{b}$, we
used the light-meson saturation approach to determine its ratio. With the so-
determined $C_{a}$ and $C_{b}$ we calculated the binding energies (if bound
states exist) and scattering lengths of all these systems as well as their
heavy quark flavor partners.

In the $\bar{D}^{(\ast)}D^{(\ast)}$ system, we obtained a $J^{PC}=2^{++}$
$\bar{D}^{\ast}D^{\ast}$ molecule with a mass of $4013$ MeV referred to as
$X(4013)$, which can be regarded as the heavy quark spin symmetry partner of
$X(3872)$. Moreover, in the bottom sector we predicted the existence of five
molecules. We extended the $\bar{D}^{(\ast)}D^{(\ast)}$ to the
$D^{(\ast)}\Xi_{cc}^{(\ast)}$ system by HADS. Two molecules, $J^{P}=1/2^{-}$
$D^{\ast}\Xi_{cc}$ and $J^{P}=5/2^{-}$ $D^{\ast}\Xi_{cc}^{\ast}$, are
obtained. There exist a complete multiplet of hadronic molecules in the
corresponding bottom sector. Using HADS again we studied the
$\bar{\Xi}_{cc}^{(\ast)}{\Xi}_{cc}^{(\ast)}$ system, and found 7 bound states,
$J^{PC}=0^{-+}$ $\bar{\Xi}_{cc}\Xi_{cc}$, $J^{PC}=1^{--}$
$\bar{\Xi}_{cc}\Xi_{cc}$, $J^{PC}=1^{-+}$ $\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$,
$J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}^{\ast}$, $J^{PC}=2^{-+}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$ and $J^{PC}=3^{--}$
$\bar{\Xi}_{cc}^{\ast}\Xi_{cc}^{\ast}$. There also exist a complete multiplet
of hadronic molecules in the bottom sector. Among them, the $J^{PC}=0^{-+}$
$\bar{\Xi}_{cc}\Xi_{cc}$ and $J^{PC}=1^{--}$ $\bar{\Xi}_{cc}\Xi_{cc}$
molecules may contribute to $X(7200)$ recently observed by the LHCb
Collaboration, which are related to $X(3872)$ via HADS.

##  V Acknowledgments

This work is partly supported by the National Natural Science Foundation of
China under Grants No.11735003, No.11975041, and No.11961141004, and the
fundamental Research Funds for the Central Universities.

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$\mathbf{V}=\begin{pmatrix}v_{11}&\mathbf{u}^{\top}\\\
\mathbf{u}&\mathbf{V}_{22}\end{pmatrix}$

and define $\mathbf{V}_{22\cdot
1}=\mathbf{V}_{22}-\frac{1}{v_{11}}\mathbf{u}\mathbf{u}^{\top}$. We will use
the following results for the inverse-Wishart distribution (Bodnar and
Okhrin,, 2008):

  1. (i)

$\frac{1}{v_{11}}\mathbf{u}|\mathbf{V}_{22\cdot
1}\sim\mathcal{N}(\mathbf{0},\mathbf{V}_{22\cdot 1})$;

  2. (ii)

$\mathbf{V}_{22\cdot 1}\sim\mathcal{W}^{-1}(\mathbf{I}_{n-1},p)$.

Then

$\mathrm{E}\left(\sum_{j\neq
1}\frac{v_{1j}^{2}}{v_{11}^{2}}\right)=\mathrm{E}\left(\frac{\mathbf{u}^{\top}\mathbf{u}}{v_{11}^{2}}\right)=\mathrm{E}\left[\mathrm{E}\left(\frac{\mathbf{u}^{\top}\mathbf{u}}{v_{11}^{2}}\Bigg{|}\mathbf{V}_{22\cdot
1}\right)\right]=\mathrm{E}\left[{\rm tr}(\mathbf{V}_{22\cdot
1})\right]=\frac{n-1}{p-n}.$

Using (A.2), we get

$\mathrm{E}\left[{\rm tr}(\mathbf{A})\right]=\frac{n(p-1)}{p-n}.$

On the other hand, we have shown in Remark 4.2 that

$\mathrm{E}(\mathrm{df}_{\rm R})=\frac{n}{2}{\rm
tr}\left[\mathrm{E}\left(\mathbf{V}^{-1}\right)\right]+\frac{n}{2}=\frac{n(p-1)}{2(p-n-1)}.$

Thus, as $n\to\infty$, $p\to\infty$ and $p/n\to\gamma>1$,

$\mathrm{E}(\xi_{\mathbf{X}})=\mathrm{E}\left[2\,\mathrm{df}_{\rm R}(S)-{\rm
tr}(\mathbf{A})\right]=\frac{n(p-1)}{(p-n-1)(p-n)}\to\frac{1}{\gamma-1}.$

∎

###  A.9 Proof of Lemma 7.1

Proof. Using the singular value decomposition of
$\mathbf{X}=\mathbf{U}(\bm{\Psi},\mathbf{O})(\mathbf{V}_{1},\mathbf{V}_{2})^{\top}$,
we have
$\hat{\bm{\beta}}=\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{y}=\mathbf{V}_{1}\mathbf{\Psi}^{-1}\mathbf{U}^{\top}\mathbf{y}$
and

$\mathbf{E}^{k}=(\mathbf{I}_{p}-\alpha\mathbf{X}^{\top}\mathbf{X})^{k}=(\mathbf{V}_{1},\mathbf{V}_{2})\begin{pmatrix}(\mathbf{I}_{n}-\alpha\bm{\Psi}^{2})^{k}&\mathbf{O}\\\
\mathbf{O}&\mathbf{I}_{p-n}\end{pmatrix}\begin{pmatrix}\mathbf{V}_{1}^{\top}\\\
\mathbf{V}_{2}^{\top}\end{pmatrix}.$

Thus, for $\alpha>0$, (7.2) converges if and only if

$-1<1-\alpha\psi_{j}^{2}<1,\text{ for }j=1,\ldots,n,$

which is equivalent to

$0<\alpha<\frac{2}{\psi_{1}^{2}}=\frac{2}{\lambda_{\max}(\mathbf{X}^{\top}\mathbf{X})}.$

∎

###  A.10 Proof of Theorem 7.1

Proof. By (7.5), we have

$\mathrm{E}(\|\mathbf{h}_{\ast}\|^{2}|\mathbf{X})={\rm
tr}[(\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}+(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X})^{\top}(\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}+(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X})\bm{\Sigma}].$

Since $\bm{\beta}^{(\infty)}$ is an interpolating model, if follows from (3.5)
that

$\mathrm{df}_{\rm R}(\bm{\beta}^{(\infty)})=\frac{n}{2}+\frac{n}{2}{\rm
tr}[(\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}+(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X})^{\top}(\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}+(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X})\bm{\Sigma}].$

Note that $\mathrm{df}_{\rm R}(\hat{\bm{\beta}})=\frac{n}{2}+\frac{n}{2}{\rm
tr}[\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-2}\mathbf{X}\bm{\Sigma}]$.
Then, we have

$\displaystyle\mathrm{df}_{\rm R}(\bm{\beta}^{(\infty)})$
$\displaystyle=\mathrm{df}_{\rm R}(\hat{\bm{\beta}})+{\rm
tr}(\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{F}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma})+2{\rm
tr}\left[\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma}\right]$
$\displaystyle=\mathrm{df}_{\rm R}(\hat{\bm{\beta}})+{\rm
tr}(\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{F}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma})+2{\rm
tr}\left[\mathbf{V}_{1}\mathbf{\Psi}^{-1}\mathbf{U}^{\top}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma}\right]$
$\displaystyle=\mathrm{df}_{\rm R}(\hat{\bm{\beta}})+{\rm
tr}(\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{F}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma})+2{\rm
tr}\left[\mathbf{\Psi}^{-1}\mathbf{U}^{\top}\mathbf{F}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\Sigma}\mathbf{V}_{1}\right].$

The second term above is nonnegative for all $\bm{\Sigma}$. When
$\bm{\Sigma}=\mathbf{I}_{p}$, the third term vanishes due to
$\mathbf{V}_{2}^{\top}\mathbf{V}_{1}=\mathbf{O}$. Thus

$\mathrm{df}_{\rm R}(\bm{\beta}^{(\infty)})\geq\mathrm{df}_{\rm
R}(\hat{\bm{\beta}}).$

The equality holds if and only if
$\mathrm{span}(\mathbf{F})\subseteq\mathrm{span}(\mathbf{V}_{1})$, i.e.,
$\bm{\beta}^{(\infty)}=\hat{\bm{\beta}}$. ∎

###  A.11 Derivation of Equation (7.6)

Let $\mathbf{h}_{\ast}$ and $\mathbf{h}_{\ast}^{(\infty)}$ be the hat vectors
at $\mathbf{x}_{\ast}$ for $\hat{\bm{\beta}}$ and $\bm{\beta}^{(\infty)}$
respectively. Note that both $\hat{\bm{\beta}}$ and $\bm{\beta}^{(\infty)}$
define interpolating models. Hence,

$\Delta
B_{\mathbf{X}}(\hat{\bm{\beta}})=\mathrm{E}[(\mu_{\ast}-\mathbf{h}_{\ast}^{\top}\bm{\mu})^{2}|\mathbf{X}]\,\text{
and }\,\Delta
B_{\mathbf{X}}(\bm{\beta}^{(\infty)})=\mathrm{E}[(\mu_{\ast}-(\mathbf{h}_{\ast}^{(\infty)})^{\top}\bm{\mu})^{2}|\mathbf{X}].$

Since we assume $\bm{\mu}(\mathbf{x};\bm{\beta})=\mathbf{x}^{\top}\bm{\beta}$,
we have

$\mu_{\ast}-\mathbf{h}_{\ast}^{\top}\bm{\mu}=\mathbf{x}_{\ast}^{\top}(\mathbf{I}_{p}-\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X})\bm{\beta}=\mathbf{x}_{\ast}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta}.$

Then, using (7.5) with $\mathbf{F}=\mathbf{P}\mathbf{X}^{\top}$, we have

$\displaystyle(\mu_{\ast}-(\mathbf{h}_{\ast}^{(\infty)})^{\top}\bm{\mu})^{2}=(\mu_{\ast}-\mathbf{h}_{\ast}^{\top}\bm{\mu})^{2}$
$\displaystyle-2(\mathbf{x}_{\ast}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta})(\mathbf{x}_{\ast}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{P}\mathbf{X}^{\top}\mathbf{X}\bm{\beta})$
$\displaystyle+(\mathbf{x}_{\ast}^{\top}\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{P}\mathbf{X}^{\top}\mathbf{X}\bm{\beta})^{2}$

Under the assumption that $\mathrm{E}(\mathbf{x}_{\ast})=0$ and
$\mathrm{Var}(\mathbf{x}_{\ast})=\mathbf{I}_{p}$,

$\displaystyle\Delta B_{\mathbf{X}}(\bm{\beta}^{(\infty)})$
$\displaystyle=\Delta
B_{\mathbf{X}}(\hat{\bm{\beta}})-2(\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta})^{\top}(\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{P}\mathbf{X}^{\top}\mathbf{X}\bm{\beta})+\|\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{P}\mathbf{X}^{\top}\mathbf{X}\bm{\beta}\|^{2}$
$\displaystyle=\Delta
B_{\mathbf{X}}(\hat{\bm{\beta}})-\|\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta}\|^{2}+\|\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta}-\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{P}\mathbf{X}^{\top}\mathbf{X}\bm{\beta}\|^{2}$
$\displaystyle=\Delta
B_{\mathbf{X}}(\hat{\bm{\beta}})-\|\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\bm{\beta}\|^{2}+\|\mathbf{V}_{2}\mathbf{V}_{2}^{\top}\mathbf{z}\|^{2}.$

##  Appendix B Examples

###  B.1 Example 3.2

Using the expression for $\mathbf{h}_{\ast}$ in the example, we have

$\|\mathbf{h}_{\ast}\|^{2}=\begin{cases}1,&a\leq x_{\ast}<x_{1}\text{ or
}x_{n}\leq x_{\ast}\leq b,\\\
K^{2}(z_{\ast,i})+(1-K(z_{\ast,i}))^{2},&x_{i}\leq
x_{\ast}<x_{i+1}.\end{cases}$

Define $C_{K}=\int_{0}^{1}\left[K^{2}(z)+(1-K(z))^{2}\right]dz$. For any
distribution on $[a,b]$ with continuous and positive density $f$, we have

$\mathrm{E}(\|\mathbf{h}_{\ast}\|^{2}\|\mathbf{X})=\int_{a}^{x_{1}}f(x_{\ast})dx_{\ast}+\sum_{i=1}^{n-1}\int_{x_{i}}^{x_{i+1}}\|\mathbf{h}_{\ast}\|^{2}f(x_{\ast})dx_{\ast}+\int_{x_{n}}^{b}f(x_{\ast})dx_{\ast}.$
(B.1)

By the mean value theorem for definite integrals, there must exist some
$\tilde{x}_{i}\in(x_{i},x_{i+1})$ such that

$\displaystyle\int_{x_{i}}^{x_{i+1}}\|\mathbf{h}_{\ast}\|^{2}f(x_{\ast})dx_{\ast}$
$\displaystyle=(x_{i+1}-x_{i})f(\tilde{x}_{i})\int_{x_{i}}^{x_{i+1}}\|\mathbf{h}_{\ast}\|^{2}\frac{1}{x_{i+1}-x_{i}}dx_{\ast}$
$\displaystyle=C_{K}(x_{i+1}-x_{i})f(\tilde{x}_{i}).$

Note that $x_{1}\xrightarrow{\rm P}a$, $x_{n}\xrightarrow{\rm P}b$ as
$n\to\infty$. By applying the above result to (B.1) and using the definition
of definite integrals as the limit of a Riemann sum, we have

$\displaystyle\mathrm{E}(\|\mathbf{h}_{\ast}\|^{2}\|\mathbf{X})$
$\displaystyle=\int_{a}^{x_{1}}f(x_{\ast})dx_{\ast}+C_{K}\sum_{i=1}^{n-1}(x_{i+1}-x_{i})f(\tilde{x}_{i})+\int_{x_{n}}^{b}f(x_{\ast})dx_{\ast}$
$\displaystyle\xrightarrow{\rm
P}C_{K}\int_{a}^{b}f(x_{\ast})dx_{\ast}=C_{K},\quad\text{as $n\to\infty$}.$

Thus,

$\frac{1}{n}\mathrm{df}_{\rm
R}=\frac{1}{2}+\frac{1}{2}\mathrm{E}(\|\mathbf{h}_{\ast}\|^{2}\|\mathbf{X})\xrightarrow{\rm
P}\frac{C_{K}+1}{2}.$

###  B.2 Example 3.3

Let $\phi_{t}(x)=x^{t}/t!$ for $t=0,1,2,\ldots$ and

$R(u,v)=\int_{0}^{1}\frac{(u-z)^{s-1}_{+}}{(s-1)!}\frac{(v-z)^{s-1}_{+}}{(s-1)!}dz,\;\text{for
}u,v\in[0,1],$

where $(\cdot)_{+}=\max(0,\cdot)$. Then, the interpolating polynomial spline
of degree $2s-1$ is of the form

$\hat{\mu}(x_{\ast})=\sum_{t=0}^{s-1}d_{t}\phi_{t}(x_{\ast})+\sum_{i=1}^{n}c_{i}R(x_{\ast},x_{i}).$
(B.2)

To write out $c_{i}$ and $d_{t}$, let $\mathbf{c}=(c_{1},\ldots,c_{n})^{\top}$
and $\mathbf{d}=(d_{0},\ldots,d_{s-1})^{\top}$. Let
$\mathbf{S}=(\phi_{t}(x_{i}))\in\mathbb{R}^{n\times s}$ and
$\mathbf{T}=(R(x_{i},x_{j}))\in\mathbb{R}^{n\times n}$. Assume $\mathbf{S}$
has Q-R decomposition

$\mathbf{S}=(\mathbf{Q}_{1},\mathbf{Q}_{2})\begin{pmatrix}\mathbf{R}\\\
\mathbf{O}\end{pmatrix},$

where $\mathbf{Q}_{1}\in\mathbb{R}^{n\times s}$,
$\mathbf{Q}_{2}\in\mathbb{R}^{n\times(n-s)}$, and
$\mathbf{R}\in\mathbb{R}^{s\times s}$ is an upper triangular matrix. Define
$\mathbf{U}=\mathbf{Q}_{2}(\mathbf{Q}_{2}^{\top}\mathbf{T}\mathbf{Q}_{2})^{-1}\mathbf{Q}_{2}^{\top}$
and
$\mathbf{V}=\mathbf{R}^{-1}\mathbf{Q}_{1}^{\top}(\mathbf{I}-\mathbf{T}\mathbf{U})$.
Then, we have

$\mathbf{c}=\mathbf{U}\mathbf{y}\;\text{ and
}\;\mathbf{d}=\mathbf{V}\mathbf{y}.$ (B.3)

For the derivation of (B.2) and (B.3), see Gu, (2013). For any
$x_{\ast}\in[0,1]$, let
$\bm{\phi}_{\ast}=(\phi_{0}(x_{\ast}),\ldots,\phi_{s-1}(x_{\ast}))^{\top}$ and
$\bm{\rho}_{\ast}=(R(x_{\ast},x_{1}),\ldots,R(x_{\ast},x_{n}))^{\top}$. Then,
we can rewrite (B.2) as

$\hat{\mu}(x_{\ast})=\bm{\phi}_{\ast}^{\top}\mathbf{d}+\bm{\rho}_{\ast}^{\top}\mathbf{c}=(\mathbf{V}^{\top}\bm{\phi}_{\ast}+\mathbf{U}^{\top}\bm{\rho}_{\ast})^{\top}\mathbf{y}.$

Since $\mathbf{U},\mathbf{V}$, $\bm{\phi}_{\ast}$ and $\bm{\rho}_{\ast}$ don’t
depend on $\mathbf{y}$, we have

$\mathbf{h}_{\ast}=\mathbf{V}^{\top}\bm{\phi}_{\ast}+\mathbf{U}^{\top}\bm{\rho}_{\ast}.$

###  B.3 Example 3.4

Assume $\frac{1}{2}\bar{L}\leq\omega\leq\underaccent{\bar}{L}$. Define
$x_{0}=a-\omega$ and $x_{n+1}=b+\omega$. By construction of the smoother, we
have

$\mathbf{h}_{\ast}=\begin{cases}\mathbf{e}_{i},&x_{i-1}+\omega<x_{\ast}<x_{i+1}-\omega,\;i=1,\ldots,n\\\
\frac{1}{2}\mathbf{e}_{i}+\frac{1}{2}\mathbf{e}_{i+1},&x_{i+1}-\omega\leq
x_{\ast}\leq x_{i}+\omega,\;i=1,\ldots,n-1.\end{cases}$

Since $\mathbf{e}_{i}\in\mathbb{R}^{n}$ is the $i$th standard basis vector,

$\|\mathbf{h}_{\ast}\|^{2}=\begin{cases}1,&x_{i-1}+\omega<x_{\ast}<x_{i+1}-\omega,\;i=1,\ldots,n\\\
\frac{1}{2},&x_{i+1}-\omega\leq x_{\ast}\leq
x_{i}+\omega,\;i=1,\ldots,n-1.\end{cases}$

When $x_{\ast}\sim\mathrm{Uniform}(a,b)$,

$\displaystyle\mathrm{E}(\|\mathbf{h}_{\ast}\|^{2}|\mathbf{X})$
$\displaystyle=\frac{1}{b-a}\sum_{i=1}^{n}(x_{i+1}-x_{i-1}-2\omega)+\frac{1}{2(b-a)}\sum_{i=1}^{n-1}(x_{i}-x_{i+1}+2\omega)$
$\displaystyle=1+\frac{x_{n}-x_{1}}{2(b-a)}-\frac{n-1}{b-a}\omega.$

Thus, by (3.5), we have

$\mathrm{df}_{\rm
R}(\omega)=n+\frac{n(x_{n}-x_{1})}{4(b-a)}-\frac{n(n-1)}{2(b-a)}\omega.$

###  B.4 Optimal Model Size in Figure 7

To study the difference of $C_{p}$ and $\widetilde{\mathrm{ErrR}}$ in model
selection, it is more convenient to look at their expected versions
$\mathrm{E}_{\mathbf{X}}(C_{p})=\mathrm{ErrF}$ and
$\mathrm{E}_{\mathbf{X}}(\widetilde{\mathrm{ErrR}})=\mathrm{ErrR}$. Assume
$\sigma_{\mathcal{S}}^{2}(p)=\alpha\left(1-\frac{p}{d}\right)^{\eta}$, where
$\alpha>0$, $\eta\geq 1$ and $d\leq n$. Here, $\eta$ is a parameter that
controls the sparsity of the true model. Implications of this assumption
include

  1. i.

The signal-to-noise ratio is $\frac{\alpha}{\sigma_{\varepsilon}^{2}}$.

  2. ii.

When $\eta=1$, $\sigma_{\mathcal{S}}^{2}$ decreases linearly in $p$, which
mimics the random selection procedure or the situation where all variables
have the same degree of importance.

  3. iii.

When $\eta>1$, $\sigma_{\mathcal{S}}^{2}$ decays faster as $p$ increases. This
portrays the situation where variables are included “presciently” from the
most important to the least.

For simplicity, consider the case when $d=n$ and $\sigma_{\varepsilon}^{2}=1$.
Let $\gamma=\frac{p}{n}$ and $c=\alpha(\eta+1)$. Here, we allow $\gamma$ to
take any real value between 0 and 1. Then, by Proposition 5.1 and (4.5), we
have

$\mathrm{ErrT}=1-\gamma+\alpha(1-\gamma)^{\eta+1}\text{ and }\mathrm{E}(\Delta
B_{\mathbf{X}})=\alpha\gamma(1-\gamma)^{\eta}+\alpha\gamma(1-\gamma)^{\eta}\frac{1}{1-\gamma-\frac{1}{n}}.$

Thus,

$\displaystyle\mathrm{ErrF}$
$\displaystyle=\mathrm{ErrT}+\frac{2}{n}\mathrm{df}_{\rm
F}=1+\gamma+\alpha(1-\gamma)^{\eta+1},$ $\displaystyle\mathrm{ErrR}$
$\displaystyle=\mathrm{ErrT}+\mathrm{E}(\Delta
B_{\mathbf{X}})+\frac{2}{n}\mathrm{E}(\mathrm{df}_{\rm R})$
$\displaystyle=1+\alpha(1-\gamma)^{\eta+1}+\alpha\gamma(1-\gamma)^{\eta}+\alpha\gamma(1-\gamma)^{\eta}\frac{1}{1-\gamma-\frac{1}{n}}+\frac{\gamma}{1-\gamma-\frac{1}{n}}.$

Assume $0\leq\gamma<1-\frac{1}{n}$. Define $\gamma_{\rm
F}^{\ast}=\operatorname*{arg\,min}_{\gamma}\mathrm{ErrF}=\max(0,1-c^{-1/\eta})$
and $\gamma_{\rm R}^{\ast}=\operatorname*{arg\,min}_{\gamma}\mathrm{ErrR}$.
Then,

$\displaystyle\frac{d\mathrm{ErrR}}{d\gamma}$
$\displaystyle=-\alpha\eta(1-\gamma)^{\eta-1}\frac{1-\frac{1}{n}}{1-\gamma-\frac{1}{n}}+\alpha(1-\gamma)^{\eta}\frac{1-\frac{1}{n}}{\left(1-\gamma-\frac{1}{n}\right)^{2}}+\frac{1-\frac{1}{n}}{\left(1-\gamma-\frac{1}{n}\right)^{2}}$
$\displaystyle>-\alpha\eta(1-\gamma)^{\eta-1}\frac{1-\frac{1}{n}}{1-\gamma-\frac{1}{n}}+\alpha(1-\gamma)^{\eta}\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}+\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}$
$\displaystyle=\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}\left[1-\alpha(\eta-1)(1-\gamma)^{\eta}\right]$
$\displaystyle>\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}\left[1-c(1-\gamma)^{\eta}\right].$

When $\gamma=\gamma_{\rm F}^{\ast}$,

$\displaystyle\frac{d\mathrm{ErrR}}{d\gamma}\Big{|}_{\gamma=\gamma_{\rm
F}^{\ast}}$ $\displaystyle>\frac{1-\frac{1}{n}}{(1-\gamma_{\rm
F}^{\ast})\left(1-\gamma_{\rm
F}^{\ast}-\frac{1}{n}\right)}[1-c(\min(1,c^{-1/\eta}))^{\eta}]$
$\displaystyle=\frac{1-\frac{1}{n}}{(1-\gamma_{\rm
F}^{\ast})\left(1-\gamma_{\rm F}^{\ast}-\frac{1}{n}\right)}[1-\min(c,c^{-1})]$
$\displaystyle\geq 0.$

Since $\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}>0$
for $\gamma<1-\frac{1}{n}$, and $1-c(1-\gamma)^{\eta}$ is an increasing
function of $\gamma$, we also have

$\frac{d\mathrm{ErrR}}{d\gamma}>\frac{1-\frac{1}{n}}{(1-\gamma)\left(1-\gamma-\frac{1}{n}\right)}\left[1-c(1-\gamma)^{\eta}\right]>0,\;\text{for
}\gamma\in\left(\gamma_{\rm F}^{\ast},1-\frac{1}{n}\right).$

This implies that $\gamma_{\rm R}^{\ast}\leq\gamma_{\rm F}^{\ast}$ for all
$\alpha>0$ and $\eta\geq 1$. In particular, when
$\frac{1}{\eta+1}\leq\alpha<\frac{1}{\eta-1}$,

$\gamma_{\rm R}^{\ast}=0\;\text{and}\;\gamma_{\rm
F}^{\ast}=1-c^{-\frac{1}{\eta}}>0.$

This means that $\mathrm{ErrR}$ favors a null model in this setting while
$\mathrm{ErrF}$ doesn’t. Such a difference could be large in terms of the
number of selected variables when $n$ is large. On the other hand, for any
fixed $\alpha>0$,

$\lim_{\eta\to\infty}\frac{d\mathrm{ErrR}}{d\gamma}\Big{|}_{\gamma=\gamma_{\rm
F}^{\ast}}=0,$

which suggests that $\gamma_{\rm R}^{\ast}$ approaches $\gamma_{\rm F}^{\ast}$
from the left as the level of sparsity increases.

###  B.5 Cancer Mortality Data Analysis

Based on the data partition of Figure 13, the first 7 and 9 variables were
respectively selected by $\widehat{\mathrm{ErrR}}_{+}$ and $\mathrm{ErrR}_{\rm
te}$. Table 4 lists the corresponding variables with their estimated
coefficients. We find the coefficient of $\log(\texttt{medIncome})$ is
counterintuitive, which has to do with its high correlation with
$\mathrm{logit}(\texttt{PctPrivateCoverage})$ ($0.817$), povertyPercent
($0.843$) and $\mathrm{logit}(\texttt{PctPublicCoverage})$ ($-0.773$).

Table 4: Estimated coefficients in the optimal models identified by
$\widehat{\mathrm{ErrR}}_{+}$ and $\mathrm{ErrR}_{\rm te}$ respectively.

Variable |  Description |  Transformation |  [2pt]Coefficient
---|---|---|---
|  [2pt]$\widehat{\mathrm{ErrR}}_{+}$ |  $\mathrm{ErrR}_{\rm te}$
PctPrivateCoverage |  Percent of county residents with private health coverage |  logit |  $-33.133$ *** |  $-31.540$ ***
incidenceRate |  Mean per capita (100,000) cancer diagnoses |  - |  $0.177$ *** |  $0.166$ ***
avgDeathRateEst2015 |  Estimated death rate based on 2015 population estimates and average number of reported cancer mortalities from 2010 to 2016 |  - |  $0.279$ *** |  $0.295$ ***
PctPublicCoverage |  Percent of county residents with government provided health coverage |  logit |  $-59.711$ *** |  $-51.907$ ***
povertyPercent |  Percent of population in poverty |  - |  $1.155$ ** |  $1.830$ **
PctUnemployed16_Over |  Percent of county residents aged 16 and over and unemployed |  logit |  $10.295$ *** |  $9.914$ ***
PctBachDeg25_Over |  Percent of county residents aged 25 and over with bachelor’s degree as the highest education attained |  logit |  $-6.502$ * |  $-5.693$
medIncome |  Median income of county residents |  log |  - |  $18.521$
PctHS18_24 |  Percent of county residents aged 18-24 with high school diploma as the highest education attained |  logit |  - |  $4.991$
|  | $\hat{\sigma}$ |  $13.639$ |  $13.538$
|  | adj.$R^{2}$ |  $0.720$ |  $0.724$

  * •

Significance codes: ‘***’ $<0.001$, ‘**’ $0.001\sim 0.01$, ‘*’ $0.01\sim
0.05$, ‘.’ $0.05\sim 0.1$, ‘ ’ $>0.1$.

In Figure 18, we illustrate the prediction results of the model identified by
$\widehat{\mathrm{ErrR}}_{+}$ on all 1,148 counties along with the population
map. We observe that large prediction errors are more likely to occur on less
populated counties. We then check the mean and standard deviation of the
prediction errors by quantiles of the county-level population in Table 5 and
find that 20% of counties with smallest population tend to be overestimated
and have much more variable prediction errors. This implies that the
assumption of constant variance in our regression model does not hold exactly.
Since death due to cancer is a relatively less likely event, we should expect
generally larger variance for the death rate of those less populous counties.

Figure 18: Prediction error (left) and population (right) maps for counties in
the focused states.

We also find a few overestimated counties in central New York, central
Pennsylvania and northeast West Virginia. Geographical factors can be one
possible explanation, as these counties are mostly situated in the Appalachian
Mountains region, which may have different natural and socioeconomic
conditions from the majority of the counties in our focused area.

Table 5: Mean and standard deviation of the prediction errors ($\hat{y}-y$) by population quantiles. Population quantile | $[0,0.2]$ | $(0.2,0.4]$ | $(0.4,0.6]$ | $(0.6,0.8]$ | $(0.8,1]$
---|---|---|---|---|---
Mean | $3.982$ | $0.077$ | $-1.711$ | $-2.154$ | $1.192$
SD | $17.143$ | $12.598$ | $12.487$ | $12.701$ | $12.224$


.

# Totally real submanifolds of $(LCS)_{n}$-Manifolds.

SHYAMAL KUMAR HUI and TANUMOY PAL

###### Abstract.

The present paper deals with the study of totally real submanifolds and
C-totally real submanifolds of $(LCS)_{n}$-manifolds with respect to Levi-
Civita connection as well as quarter symmetric metric connection. It is proved
that scalar curvature of C-totally real submanifolds of $(LCS)_{n}$-manifold
with respect to both the said connections are same.

###### Key words and phrases:

$(LCS)_{n}$-manifold, totally real submanifold, quarter symmetric metric
connection

###### 2000 Mathematics Subject Classification:

53C15, 53C25.

##  1\. Introduction

As a generalization of LP-Sasakian manifold, recently Shaikh [11] introduced
the notion of Lorentzian concircular structure manifolds (briefly,
$(LCS)_{n}$-manifolds) with an example. Such manifolds has many applications
in the general theory of relativity and cosmology ([12], [13]).
The notion of semisymmetric linear connection on a smooth manifold is
introduced by Friedmann and Schouten [3]. Then Hayden [5] introduced the idea
of metric connection with torsion on a Riemannian manifold. Thereafter Yano
[16] studied semisymmetric metric connection on a Riemannian manifold
systematically. As a generalization of semisymmetric connection, Golab [4]
introduced the idea of quarter symmetric linear connection on smooth
manifolds. A linear connection $\overline{\nabla}$ in an $n$-dimensional
smooth manifold $\widetilde{M}$ is said to be a quarter symmetric connection
[4] if its torsion tensor $T$ is of the form

$\displaystyle T(X,Y)$ $\displaystyle=$
$\displaystyle\overline{\nabla}_{X}Y-\overline{\nabla}_{Y}X-[X,Y]$
$\displaystyle=\eta(Y)\phi X-\eta(X)\phi Y,$

where $\eta$ is an 1-form and $\phi$ is a tensor of type $(1,1)$. In
particular, if $\phi X=X$ then the quarter symmetric connection reduces to
semisymmetric connection. Further if the quarter symmetric connection
$\overline{\nabla}$ satisfies the condition $(\overline{\nabla}_{X}g)(Y,Z)=0$
for all $X,\ Y,\ Z\in\chi(\widetilde{M})$, the set of all smooth vector fields
on $\widetilde{M}$, then $\overline{\nabla}$ is said to be a quarter symmetric
metric connection.
Due to important applications in applied mathematics and theoretical physics,
the geometry of submanifolds has become a subject of growing interest.
Analogous to almost Hermitian manifolds, the invariant and anti-invariant
submanifolds are depend on the behaviour of almost contact metric structure
$\phi$. A submanifold of a contact metric manifold $\widetilde{M}$ is said to
be anti-invariant if for any $X$ tangent to $M$, $\phi X$ is normal to $M$,
i.e., $\phi(TM)\subset T^{\bot}M$ at every point of $M$, where $T^{\bot}M$ is
the normal bundle of $M$. So, if a submanifold $M$ of a contact metric
manifold $\widetilde{M}$ is normal to the structure vector field $\xi$, then
it is anti-invariant. A submanifold $M$ in a contact metric manifold
$\widetilde{M}$ is called a C-totally real submanifold if every tangent vector
of $M$ belongs to the contact distribution [15]. Thus a submanifold $M$ in a
contact metric manifold is a C-totally real submanifold if $\xi$ is normal to
$M$. Consequently C-totally real submanifolds in a contact metric manifold are
anti-invariant, as they are normal to $\xi$. Recently Hui et al. ([1], [6],
[7], [14]) studied submanifolds of $(LCS)_{n}$-manifolds. The present paper
deals with the totally real submanifolds and C-totally real submanifolds of
$(LCS)_{n}$-manifolds with respect to Levi-Civita as well as quarter symmetric
metric connection. It is shown that the scalar curvature of a C-totally real
submanifold of $(LCS)_{n}$-manifolds with respect to Levi-Civita connection
and quarter symmetric metric connection are same. However in case of totally
real submanifolds of $(LCS)_{n}$-manifolds with respect to Levi-Civita
connection and quarter symmetric metric connection they are different. An
inequality for the square length of the shape operator in case of totally real
submanifold of $(LCS)_{n}$-manifold is derived. The equality case is also
considered.

##  2\. preliminaries

Let $\widetilde{M}$ be an $n$-dimensional Lorentzian manifold [10] admitting a
unit timelike concircular vector field $\xi$, called the characteristic vector
field of the manifold. Then we have

(2.1) $g(\xi,\xi)=-1.$

Since $\xi$ is a unit concircular vector field, it follows that there exists a
non-zero 1-form $\eta$ such that for

(2.2) $g(X,\xi)=\eta(X),$

the equation of the following form holds [17]

(2.3) $(\widetilde{\nabla}_{X}\eta)(Y)=\alpha\\{g(X,Y)+\eta(X)\eta(Y)\\},\ \ \
\alpha\neq 0,$ (2.4) $\widetilde{\nabla}_{X}\xi=\alpha\\{X+\eta(X)\xi\\},\ \ \
\alpha\neq 0,$

for all vector fields $X$, $Y$, where $\widetilde{\nabla}$ denotes the
operator of covariant differentiation with respect to the Lorentzian metric
$g$ and $\alpha$ is a non-zero scalar function satisfies

(2.5) ${\widetilde{\nabla}}_{X}\alpha=(X\alpha)=d\alpha(X)=\rho\eta(X),$

$\rho$ being a certain scalar function given by $\rho=-(\xi\alpha)$. Let us
take

(2.6) $\phi X=\frac{1}{\alpha}\widetilde{\nabla}_{X}\xi,$

then from (2.4) and (2.6) we have

(2.7) $\phi X=X+\eta(X)\xi,$ (2.8) $g(\phi X,Y)=g(X,\phi Y),$

from which it follows that $\phi$ is a symmetric (1,1) tensor and called the
structure tensor of the manifold. Thus the Lorentzian manifold $\widetilde{M}$
together with the unit timelike concircular vector field $\xi$, its associated
1-form $\eta$ and an (1,1) tensor field $\phi$ is said to be a Lorentzian
concircular structure manifold (briefly, $(LCS)_{n}$-manifold), [11].
Especially, if we take $\alpha=1$, then we can obtain the LP-Sasakian
structure of Matsumoto [9]. In a $(LCS)_{n}$-manifold $(n>2)$, the following
relations hold [11]:

(2.9) $\eta(\xi)=-1,\ \ \phi\xi=0,\ \ \ \eta(\phi X)=0,\ \ \ g(\phi X,\phi
Y)=g(X,Y)+\eta(X)\eta(Y),$ (2.10) $\phi^{2}X=X+\eta(X)\xi,$ (2.11)
$\widetilde{R}(X,Y)Z=\phi\widetilde{R}(X,Y)Z+(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\xi$

for all $X,\ Y,\ Z\in\Gamma(T\widetilde{M})$ Using (2.8) in (2.11) we get,

(2.12) $\widetilde{R}(X,Y,Z,W)=\widetilde{R}(X,Y,Z,\phi
W)+(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\eta(W)$

Let $M$ be a submanifold of dimension $m$ of a $(LCS)_{n}$-manifold
$\widetilde{M}$ $(m<n)$ with induced metric $g$. Also let $\nabla$ and
$\nabla^{\perp}$ be the induced connection on the tangent bundle $TM$ and the
normal bundle $T^{\perp}M$ of $M$ respectively. Then the Gauss and Weingarten
formulae are given by

(2.13) $\widetilde{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)$

and

(2.14) $\widetilde{\nabla}_{X}V=-A_{V}X+\nabla^{\perp}_{X}V$

for all $X,\ Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and
$A_{V}$ are second fundamental form and the shape operator (corresponding to
the normal vector field $V$) respectively for the immersion of $M$ into
$\widetilde{M}$ and they are related by [18]

(2.15) $g(h(X,Y),V)=g(A_{V}X,Y)$

for any $X,\ Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$. And the equation of
Gauss is given by

(2.16) $\widetilde{R}(X,Y,Z,W)=R(X,Y,Z,W)+g(h(X,Z),h(Y,W))-g(h(X,W),h(Y,Z))$

for any vectors $X,\ Y,\ Z,\ W$ tangent to $M$.
Let $\\{e_{i}:i=1,2,\cdots,n\\}$ be an orthonormal basis of the tangent space
$\widetilde{M}$ such that, refracting to $M^{m}$,
$\\{e_{1},e_{2},\cdots,e_{m}\\}$ is the orthonormal basis to the tangent space
$T_{x}M$ with respect to induced connection.
We write

$h_{ij}^{r}=g(h(e_{i},e_{j}),e_{r}).$

Then the square length of $h$ is

$\lvert\lvert
h\rvert\rvert^{2}=\sum_{i,j=1}^{m}g(h(e_{i},e_{j}),h(e_{i},e_{j}))$

and the mean curvature of $M$ associated to $\nabla$ is

$H=\frac{1}{m}\sum_{i=1}^{m}h(e_{i},e_{i}).$

The quarter symmetric metric connection $\overline{\widetilde{\nabla}}$ and
Riemannian connection $\widetilde{\nabla}$ on a $(LCS)_{n}$-manifold
$\widetilde{M}$ are related by [8]

(2.17) $\overline{\widetilde{\nabla}}_{X}Y=\widetilde{\nabla}_{X}Y+\eta(Y)\phi
X-g(\phi X,Y)\xi.$

If $\overline{\widetilde{R}}$ and $\widetilde{R}$ are the curvature tensors of
a $(LCS)_{n}$-manifold $\widetilde{M}$ with respect to quarter symmetric
metric connection $\overline{\widetilde{\nabla}}$ and Riemannian connection
$\widetilde{\nabla}$, then

$\displaystyle\overline{\widetilde{R}}(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle\widetilde{R}(X,Y,Z,W)+(2\alpha-1)[g(\phi X,Z)g(\phi Y,W)$
$\displaystyle-g(\phi Y,Z)g(\phi X,W)]+\alpha[\eta(Y)g(X,W)$
$\displaystyle-\eta(X)g(Y,W)]\eta(Z)+\alpha[g(Y,Z)\eta(X)$
$\displaystyle-g(X,Z)\eta(Y)]\eta(W)$

for all $X,\ Y,\ Z\ \text{and}\ W$ on $\chi(\widetilde{M})$.
Let $L$ be a $k$-plane section of $T_{x}M$ and $X$ be a unit vector in $L$. We
choose an orthonormal basis $\\{e_{1},e_{2},\cdots,e_{k}\\}$ of $L$ such that
$e_{1}=X$. Then the Ricci curvature $Ric_{L}$ of $L$ at $X$ is defined by

(2.19) $Ric_{L}(X)=K_{12}+K_{13}+\cdots+K_{1k},$

where $K_{ij}$ denotes the sectional curvature of the $2$-plane section
spanned by $e_{i},e_{j}$. Such a curvature is called a $k$-Ricci curvature.
The scalar curvature $\tau$ of the $k$-plane section $L$ is given by

(2.20) $\tau(L)=\displaystyle\sum_{i\leq i<j\leq k}K_{ij}.$

For each integer $k$, $2\leq k\leq n$, the Riemannian invariant $\Theta_{k}$
on an $n$-dimensional Riemannian manifold $M$ is defined by

(2.21) $\Theta_{k}(x)=\frac{1}{k-1}\displaystyle\inf_{L.X}Ric_{L}(X),\ \ \ \
x\in M,$

where $L$ runs over all $k$-plane sections in $T_{x}M$ and $X$ runs over all
unit vectors in $L$. indent The relative null space for a submanifold $M$ of a
Riemannian manifold at a point $x\in M$ is defined by

(2.22) $\emph{N}_{x}=\\{X\in T_{x}M|h(X,Y)=0,Y\in T_{x}M\\}.$

##  3\. Main results

This section deals with the study of totally real submanifolds of
$(LCS)_{n}$-manifolds with respect to Levi-Civita as well as quarter symmetric
metric connection. We prove the following:

######  Theorem 3.1.

Let $M$ be a toatally real submanifold of dimension $m$ $(m<n)$ of a
$(LCS)_{n}$-manifold $\widetilde{M}$. Then

(3.1) $m^{2}\lvert\lvert H\rvert\rvert^{2}=2\tau+\lvert\lvert
h\rvert\rvert^{2}+(m-1)(\alpha^{2}-\rho),$

where $\tau$ is the scalar curvature of $M$.

###### Proof.

Let $M$ be a totally real submanifold of a $(LCS)_{n}$-manifold
$\widetilde{M}$. Now from (2.12) and (2.16), we get

$\displaystyle R(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle\widetilde{R}(X,Y,Z,\phi W)+(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)$
$\displaystyle-g(X,Z)\eta(Y)\\}\eta(W)+g(h(X,W),h(Y,Z))$
$\displaystyle-g(h(X,Z),h(Y,W))$

for any $X,\ Y,\ Z,\ W\in\Gamma(TM)$.
Since $M$ is totally real submanifold i.e., anti-invariant so

$\widetilde{R}(X,Y,Z,\phi W)=g(\widetilde{R}(X,Y)Z,\phi W)=0$

and hence (3) yields

$\displaystyle R(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\eta(W)$
$\displaystyle+g(h(X,W),h(Y,Z))-g(h(X,Z),h(Y,W))$

for any $X,\ Y,\ Z,\ W\in\Gamma(TM)$. Putting $X=W=e_{i}$ and $Y=Z=e_{j}$ in
(3) and taking summation over $1\leq i<j\leq m$, we get

$\displaystyle\displaystyle\sum_{1\leq i<j\leq m}R(e_{i},e_{j},e_{j},e_{i})$
$\displaystyle=$ $\displaystyle(\alpha^{2}-\rho)\displaystyle\sum_{1\leq
i<j\leq
m}[g(e_{j},e_{j})\eta(e_{i})\eta(e_{i})-g(e_{i},e_{j})\eta(e_{j})\eta(_{j})]$
$\displaystyle+\displaystyle\sum_{1\leq i<j\leq
m}g(h(e_{i},e_{i}),h(e_{j},e_{j}))-\displaystyle\sum_{1\leq i<j\leq
m}g(h(e_{i},e_{j}),h(e_{j},e_{i}))$

i.e.,

(3.4) $2\tau=-(m-1)(\alpha^{2}-\rho)+m^{2}\lvert\lvert
H\rvert\rvert^{2}-\lvert\lvert h\rvert\rvert^{2},$

which implies (3.1). ∎

######  Corollary 3.1.

Let $M$ be a C-totally real submanifold of dimension $m$ $(m<n)$ of a
$(LCS)_{n}$-manifold $\widetilde{M}$. Then

$m^{2}\lvert\lvert H\rvert\rvert^{2}=2\tau+\lvert\lvert h\rvert\rvert^{2},$

where $\tau$ is the scalar curvature of $M$.

###### Proof.

In a C-totally real submanifold, it is known that $\eta(X)=0$ for all
$X\in\Gamma(TM)$ and $\xi\in T^{\bot}M$. Then (3) yields

$\displaystyle R(X,Y,Z,W)$ $\displaystyle=$ $\displaystyle
g(h(X,W),h(Y,Z))-g(h(X,Z),h(Y,W)),$

from which as similar in above we can prove that $m^{2}\lvert\lvert
H\rvert\rvert^{2}=2\tau+\lvert\lvert h\rvert\rvert^{2}$. ∎

Now let $M$ be a submanifold of dimension $m$ $(m<n)$ of a
$(LCS)_{n}$-manifold $\widetilde{M}$ with respect to quarter symmetric metric
connection $\overline{\widetilde{\nabla}}$ and $\overline{\nabla}$ be the
induced connection of $M$ associated to the quarter symmetric metric
connection. Also let $\overline{h}$ be the second fundamental form of $M$ with
respect to $\overline{\nabla}$. Then the Gauss formula can be written as

(3.5)
$\overline{\widetilde{\nabla}}_{X}Y=\overline{\nabla}_{X}Y+\overline{h}(X,Y)$

and hence by virtue of (2.13) and (2.17) we get

(3.6) $\displaystyle\overline{\nabla}_{X}Y+\overline{h}(X,Y)$ $\displaystyle=$
$\displaystyle\nabla_{X}Y+h(X,Y)+\eta(Y)\phi X-g(\phi X,Y)\xi$

If $M$ is totally real submanifold of $\widetilde{M}$ then $\phi X\in
T^{\bot}M$ for any $X\in TM$ and hence $g(\phi X,Y)=0$ for $X,\ Y\in TM$. So,
equating the normal part from (3.6) we get

(3.7) $\overline{h}(X,Y)=h(X,Y)+\eta(Y)\phi X.$

Further, if $M$ is C-totally real submanifold of $\widetilde{M}$ then $\xi\in
T^{\bot}M$ and hence $\eta(Y)=0$ for all $Y\in TM$. So, (3.7) yields

(3.8) $\overline{h}(X,Y)=h(X,Y).$

Let $U$ be a unit tangent vector at $x\in\widetilde{M}$ and
$\\{e_{i}:i=1,2,\cdots,n\\}$ be an orthonormal basis of the tangent space
$\widetilde{M}$ such that $e_{1}=U$ refracting to $M^{m}$,
$\\{e_{1},e_{2},\cdots,e_{m}\\}$ is the orthonormal basis to the tangent space
$T_{x}M$ with respect to induced quarter symmetric metric connection. Then we
have the following:

######  Theorem 3.2.

Let $M$ be a totally real submanifold of a $(LCS)_{n}$-manifold
$\widetilde{M}$ with respect to quarter symmetric metric connection then

(3.9) $m^{2}\lvert\lvert H\rvert\rvert^{2}=2\overline{\tau}+\lvert\lvert
h\rvert\rvert^{2}+(2m-1)\alpha+m\alpha\eta^{2}(U),$

where $\overline{\tau}$ is the scalar curvature of $M$ with respect to induced
connection associated to the quarter symmetric metric connection.

###### Proof.

In case of $(LCS)_{n}$-manifold $\widetilde{M}$ with respect to quarter
symmetric metric connection, the relation (2.16) becomes

$\displaystyle\overline{\widetilde{R}}(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle\overline{R}(X,Y,Z,W)+g(\overline{h}(X,Z),\overline{h}(Y,W))$
$\displaystyle-g(\overline{h}(X,W),\overline{h}(Y,Z)).$

In view of (2.7) and (2.8) (3) yields

$\displaystyle\overline{R}(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle\widetilde{R}(X,Y,Z,\phi W)+(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)$
$\displaystyle-g(X,Z)\eta(Y)\\}\eta(W)+(2\alpha-1)[g(\phi X,Z)g(\phi Y,W)$
$\displaystyle-g(\phi Y,Z)g(\phi X,W)]+\alpha[\eta(Y)g(X,W)$
$\displaystyle-\eta(X)g(Y,W)]\eta(Z)+\alpha[g(Y,Z)\eta(X)$
$\displaystyle-g(X,Z)\eta(Y)]\eta(W)$
$\displaystyle+g(\overline{h}(X,W),\overline{h}(Y,Z))-g(\overline{h}(X,Z),\overline{h}(Y,W)).$

Since $M$ is totally real, therefore $g(\phi X,Y)=0$ for all $X,\ Y\in TM$ and
(3.7) holds. Thus (3) becomes

$\displaystyle\overline{R}(X,Y,Z,W)$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\rho)\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\eta(W)$
$\displaystyle+\alpha[\eta(Y)g(X,W)-\eta(X)g(Y,W)]\eta(Z)$
$\displaystyle+\alpha[g(Y,Z)\eta(X)-g(X,Z)\eta(Y)]\eta(W)$
$\displaystyle+g(h(X,W),h(Y,Z))-g(h(X,Z),h(Y,W))$
$\displaystyle-\eta(Z)g(h(X,W),\phi Y)-\eta(W)g(\phi X,h(Y,Z))$
$\displaystyle+\eta(Z)g(\phi X,h(Y,W))+\eta(W)g(h(X,Z),\phi Y).$

Putting $X=W=e_{i}$ and $Y=Z=e_{j}$ in (3) and taking summation over $1\leq
i<j\leq m$ we get

$\displaystyle 2\overline{\tau}$ $\displaystyle=$
$\displaystyle-(m-1)(\alpha^{2}-\rho)-\alpha(1+\eta^{2}(U))m-\alpha(m-1)$
$\displaystyle+m^{2}\lvert\lvert H\rvert\rvert^{2}-\lvert\lvert
h\rvert\rvert^{2},$

from which (3.9) follows. ∎

######  Corollary 3.2.

Let $M$ be a C-totally real submanifold of a $(LCS)_{n}$-manifold
$\widetilde{M}$ with respect to quarter symmetric metric connection. Then

(3.14) $m^{2}\lvert\lvert H\rvert\rvert^{2}=2\overline{\tau}+\lvert\lvert
h\rvert\rvert^{2},$

where $\overline{\tau}$ is the scalar curvature of $M$ with respect to induced
quarter symmetric metric connection.

###### Proof.

If $M$ is C-totally real submanifold then $\eta(Y)=0$ for all $Y\in TM$ and
hence (3) implies that

(3.15) $\overline{R}(X,Y,Z,W)=g(h(X,Z),h(Y,W))-g(h(X,W),h(Y,Z))$

from which, by similar as above (3.14) follows. ∎

From Corollary 3.1 and Corollary 3.2 we get $\tau=\overline{\tau}$ i.e., the
scalar curvature of C-totally real submanifold of a $(LCS)_{n}$-manifold with
respect to induced Levi-Civita connection and induced quarter symmetric metric
connection are identical. Thus we can state the following:

######  Theorem 3.3.

Let $M$ be a C-totally real submanifold of a $(LCS)_{n}$-manifold
$\widetilde{M}$. Then the scalar curvature of $M$ with respect to induced
Levi-Civita connection and induced quarter symmetric metric connection are
same.

Next we prove the following:

######  Theorem 3.4.

Let $\widetilde{M}$ be a $(LCS)_{n}$-manifold and $M$ be a totally real
submanifold of $\widetilde{M}$ of dimension $m\ (m<n)$. Then
(i) for each unit vector $X\in T_{x}M$,

(3.16) $4Ric(X)\leq m^{2}\lvert\lvert
H\rvert\rvert^{2}+2(\alpha^{2}-\rho)(m-2)+4(m-2)(\alpha^{2}-\rho)\eta^{2}(X);$

(ii) in case of H(x)=0, a unit tangent vector $X$ at $x$ satisfies the
equality case of (3.16) if and only if $X$ lies in the relative null space
$\emph{N}_{x}$ at $x$.
(iii) the equality case of (3.16) holds identically for all unit tangent
vectors at $x$ if and only if either $x$ is a totally geodesic point or $m=2$
and $x$ is a totally umbilical point.

###### Proof.

Let $X\in T_{x}M$ be a unit tangent vector at $x$. We choose an orthonormal
basis $\\{e_{1},e_{2},\cdots,e_{m},e_{m+1},\cdots,e_{n}\\}$ such that
$\\{e_{1},\cdots,e_{m}\\}$ are tangent to $M$ at $x$ and $e_{1}=X$. Then from
(3.1) we have

$\displaystyle m^{2}\|H\|^{2}$ $\displaystyle=$ $\displaystyle
2\tau+\displaystyle\sum_{r=m+1}^{n}\\{(h_{11}^{r})^{2}+(h_{22}^{r}+\cdots+h_{mm}^{r})^{2})\\}$
$\displaystyle-2\displaystyle\sum_{r=m+1}^{n}\sum_{21\leq i<j\leq
n}h_{ii}^{r}h_{jj}^{r}+(m-1)(\alpha^{2}-\rho)$ $\displaystyle=$ $\displaystyle
2\tau+\frac{1}{2}\displaystyle\sum_{r=m+1}^{n}\\{(h_{11}^{r}+\cdots+h_{mm}^{r})^{2}+(h_{11}^{r}-h_{22}^{r}-\cdots-
h_{mm}^{r})^{2}\\}$
$\displaystyle+2\displaystyle\sum_{r=m+1}^{n}\sum_{i<j}(h_{ij}^{r})^{2}-2\displaystyle\sum_{r=m+1}^{n}\sum_{2\leq
i<j\leq n}h_{ii}^{r}h_{jj}^{r}+(m-1)(\alpha^{2}-\rho).$

From the equation of Gauss, we find

$K_{ij}=\displaystyle\sum_{r=m+1}^{n}[h_{ii}^{r}h_{jj}^{r}-(h_{ij}^{r})^{2}]+(\alpha^{2}-\rho)\eta^{2}(e_{i}),$

and consequently

(3.18) $\displaystyle\displaystyle\sum_{2\leq i<j\leq
m}K_{ij}=\displaystyle\sum_{r=m+1}^{n}\sum_{2\leq i<j\leq
m}[h_{ii}^{r}h_{jj}^{r}-(h_{ij}^{r})^{2}]+(\alpha^{2}-\rho)[m-2+\eta^{2}(X)].$

Using (3.18) in (3) we get

$\displaystyle m^{2}\lvert\lvert H\rvert\rvert^{2}$ $\displaystyle\geq$
$\displaystyle 2\tau+\frac{m^{2}}{2}\lvert\lvert
H\rvert\rvert^{2}+2\displaystyle\sum_{r=m+1}^{n}\sum_{j=2}^{m}(h_{1j}^{r})^{2}-2\displaystyle\sum_{2\leq
i<j\leq m}K_{ij}$
$\displaystyle-(\alpha^{2}-\rho)(m-3)-2(m-2)(\alpha^{2}-\rho)\eta^{2}(X).$

Therefore,

$m^{2}\lvert\lvert H\rvert\rvert^{2}\frac{1}{2}\geq
2Ric(X)-(\alpha^{2}-\rho)(m-3)-2(m-2)(\alpha^{2}-\rho)\eta^{2}(X),$

from which we get (3.16)
Let us assume that $H(x)=0$. Then the equality holds in (3.16) if and only if

$h_{11}^{r}=h_{22}^{r}=\cdots=h_{1m}^{r}=0,\text{and}\ \ \
h_{11}^{r}=h_{22}^{r}+\cdots+h_{mm}^{r},\ \ \ r\in\\{m+1,\cdots,n\\}.$

Then $h_{1j}^{r}=0$ for every $j\in\\{1,\cdots m\\},r\in\\{m+1\cdots n\\}$,
i.e., $X\in\emph{N}_{x}$.
(iii) The equality case of (3.16) holds for every unit tangent vector at $x$
if and only if

$h_{ij}^{r}=0,i\neq j\ \text{and}\
h_{11}^{r}+h_{22}^{r}+\cdots+h_{mm}^{r}-2h_{ii}^{r}=0.$

We distinguish two cases:
(a) $m\neq 2$,then $x$ is a totally geodesic point;
(b)$m=2$, it follows that $x$ is a totally umbilical point.
The converse is trivial. ∎

Next we obtain the following:

######  Theorem 3.5.

Let $M$ be a totally real submanifold of dimension $m$ of a $(LCS)_{n}$
manifold $\widetilde{M}(m<n)$ . Then we have

(3.20) $\displaystyle\lvert\lvert
H\rvert\rvert^{2}\geq\frac{2\tau}{m(m-1)}+\frac{1}{m}(\alpha^{2}-\rho).$

###### Proof.

We choose an orthonormal basis $\\{e_{1},\cdots e_{m},e_{m+1},\cdots e_{n}\\}$
at $x$ such that $e_{m+1}$ is parallel to the mean curvature vector $H(x)$,
and $e_{1},\cdots e_{m}$ diagonalise the shape operator $A_{e_{m+1}}$. Then
the shape operator takes the form

(3.25) $\displaystyle
A_{e_{m+1}}=\left(\begin{array}[]{ccccc}a_{1}&0&0&\cdots&0\\\
0&a_{2}&0&\cdots&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&a_{n}\\\ \end{array}\right),$

$A_{e_{r}}=(h_{ij}^{r}),\ \ i,j=1,\cdots,m;r=m+2,\cdots,n,\ \ \ \ \
traceA_{e_{r}}=\displaystyle\sum_{i=1}^{m}h_{ii}^{r}=0$
and from (3.1) we get

(3.26) $m^{2}\lvert\lvert
H\rvert\rvert^{2}=2\tau+\displaystyle\sum_{i=1}^{m}a_{i}^{2}+\displaystyle\sum_{r=m+2}^{n}\sum_{i,j=1}^{m}(h_{ij}^{r})^{2}+(m-1)(\alpha^{2}-\rho).$

On the other hand, since

(3.27)
$0\leq\displaystyle\sum_{i<j}(a_{i}-a_{j})^{2}=(m-1)\displaystyle\sum_{i}a_{i}^{2}-2\displaystyle\sum_{i<j}a_{i}a_{j},$

we obtain

(3.28) $m^{2}\lvert\lvert
H\rvert\rvert^{2}=\left(\displaystyle\sum_{i=1}^{m}a_{i}\right)^{2}+2\displaystyle\sum_{i}a_{i}^{2}-2\displaystyle\sum_{i<j}a_{i}a_{j}\leq
m\displaystyle\sum_{i=1}^{m}a_{i}^{2},$

which implies that

(3.29) $\displaystyle\sum_{i}a_{i}^{2}\geq m\|H\|^{2}.$

In view of (3.29), (3.26) yields

(3.30) $m^{2}\|H\|^{2}\geq 2\tau+m\|H\|^{2}+(m-1)(\alpha^{2}-\rho),$

which implies (3.20). ∎

######  Theorem 3.6.

Let $M$ be a totally real submanifold of dimension $m$ of a $(LCS)_{n}$
manifold $\widetilde{M}(m<n)$ . Then for any integer $k$, $2\leq k\leq m$, and
any point $x\in M$, we have

(3.31) $\|H\|^{2}(x)\geq\Theta_{k}(x)+\frac{1}{m}(\alpha^{2}-\rho).$

###### Proof.

Let $\\{e_{1},e_{2},\cdots e_{m}\\}$ be an orthonormal basis of $T_{x}M$.
Denote by $L_{i_{1},\cdots i_{k}}$ the $k$-plane section spanned by
$e_{i_{1}},\cdots e_{i_{k}}$. Then we have [2]

(3.32) $\tau(x)\geq\frac{m(m-1)}{2}\Theta_{k}(x).$

Using (3.32) in (3.20), (3.31) follows. ∎

Acknowledgement: The first author (S. K. Hui) gratefully acknowledges to the
SERB (Project No.: EMR/2015/002302), Govt. of India for financial assistance
of the work.

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S. K. Hui
Department of Mathematics, The University of Burdwan, Golapbag, Burdwan –
713104, West Bengal, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

T. Pal
A. M. J. High School, Mankhamar, Bankura – 722144, West Bengal, India
E-mail<EMAIL_ADDRESS>