# Study on the mixing of $\Xi_{c}$ and $\Xi^{\prime}_{c}$ by the transition
$\Xi_{b}\to\Xi^{(^{\prime})}_{c}$

Hong-Wei<EMAIL_ADDRESS>Gang-Yang Fang1 and Yan-Liang
<EMAIL_ADDRESS>1 School of Science, Tianjin University, Tianjin
300072, China
2 Princeton Neuroscience Institute, Princeton University, Princeton, NJ 08544,
USA

###### Abstract

Recently, the LHCb collaboration has observed the decays
$\Xi^{0}_{b}\to\Xi^{+}_{c}D^{-}_{s}$ and $\Xi^{-}_{b}\to\Xi^{0}_{c}D^{-}_{s}$.
They measured the relative branching fractions times the ratio of beauty-
baryon production cross-sections
$\mathcal{R}(\frac{\Xi^{0}_{b}}{\Lambda_{b}})\equiv\frac{\sigma(\Xi_{b}^{0})}{\sigma(\Lambda^{0}_{b})}\times\frac{B(\Xi^{0}_{b}\to\Xi^{+}_{c}D^{-}_{s})}{B(\Lambda^{0}_{b}\to\Lambda^{+}_{c}D^{-}_{s})}$
and
$\mathcal{R}(\frac{\Xi^{-}_{b}}{\Lambda_{b}})\equiv\frac{\sigma(\Xi^{-}_{b})}{\sigma(\Lambda^{0}_{b})}\times\frac{B(\Xi^{-}_{b}\to\Xi^{0}_{c}D^{-}_{s})}{B(\Lambda^{0}_{b}\to\Lambda^{+}_{c}D^{-}_{s})}$.
Once the ratio $\frac{\sigma(\Xi_{b}^{0})}{\sigma(\Lambda^{0}_{b})}$ or
$\frac{\sigma(\Xi_{b}^{-})}{\sigma(\Lambda^{0}_{b})}$ is known, one can
determine the relative branching fractions which can be used to exam the
mixing of $\Xi_{c}$ and $\Xi^{\prime}_{c}$. In previous literature, $\Xi_{c}$
and $\Xi^{\prime}_{c}$ were assumed to belong to SU(3)F antitriple and sextet,
respectively. However, recent experimental measurements such as the ratio
$\Gamma(\Xi_{cc}\to\Xi_{c}\pi^{+})/\Gamma(\Xi_{cc}\to\Xi^{\prime}_{c}\pi^{+})$
indicate the spin-flavor structures of $\Xi_{c}$ and $\Xi^{\prime}_{c}$ are a
mixture of $\Xi^{\bar{3}}_{c}$ and $\Xi^{6}_{c}$. The exact value of mixing
angle $\theta$ is still under debate. In theoretical models, the mixing angle
was fitted to be about $16.27^{\circ}\pm 2.30^{\circ}$ or $85.54^{\circ}\pm
2.30^{\circ}$ based on decay channels $\Xi_{cc}\to\Xi^{(^{\prime})}_{cc}$.
While in lattice calculation, a small angle ($1.2^{\circ}\pm 0.1^{\circ}$) is
preferred. To address such discrepancy and test the mixing of $\Xi_{c}$ and
$\Xi^{\prime}_{c}$, here we propose the analysis of semileptonic and non-
leptonic decays of $\Xi_{b}\to\Xi_{c}$ and $\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$.
We calculate the decay rate of $\Xi_{b}\to\Xi_{c}$ and
$\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$ based on light-front quark model and study
the effect of the mixing angle on the ratios of weak decays
$\Xi_{b}\to\Xi_{c}$ and $\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$. In particular, we
find the transition $\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$ can be an ideal channel
to verify the mixing and extract the mixing angle because in theory the decay
rate would be extremely tiny without mixing. Our calculation suggests a
measurement of $\Xi_{b}\to\Xi^{\prime}_{c}$ can be feasible in the near
future, which will help to test flavor mixing angle $\theta$ and elucidate the
mechanism of decay of heavy baryons.

###### pacs:

13.30.-a,12.39.Ki, 14.20.Mr

##  I Introduction

Recently the LHCb collaboration observed the decays
$\Xi^{0}_{b}\to\Xi^{+}_{c}D^{-}_{s}$ and
$\Xi^{-}_{b}\to\Xi^{0}_{c}D^{-}_{s}$LHCb:2023ngz . They measured the relative
branching fractions times the ratio of the corresponding beauty-baryon
production cross-sections
$\mathcal{R}(\frac{\Xi^{0}_{b}}{\Lambda_{b}})\equiv\frac{\sigma(\Xi_{b}^{0})}{\sigma(\Lambda^{0}_{b})}\times\frac{B(\Xi^{0}_{b}\to\Xi^{+}_{c}D^{-}_{s})}{B(\Lambda^{0}_{b}\to\Lambda^{+}_{c}D^{-}_{s})}=15.8\pm
1.1\pm 0.6\pm 7.7$ and
$\mathcal{R}(\frac{\Xi^{-}_{b}}{\Lambda_{b}})\equiv\frac{\sigma(\Xi^{-}_{b})}{\sigma(\Lambda^{0}_{b})}\times\frac{B(\Xi^{-}_{b}\to\Xi^{0}_{c}D^{-}_{s})}{B(\Lambda^{0}_{b}\to\Lambda^{+}_{c}D^{-}_{s})}=16.9\pm
1.3\pm 0.9\pm 4.3$, which suggests experimental measurement of transitions
$\Xi_{b}\to\Xi_{c}$ and $\Xi_{b}\to\Xi^{\prime}_{c}$ can be promising in the
near future. These channels can be used to probe the possible mixing of
$\Xi_{c}$ and $\Xi^{\prime}_{c}$ states.

Historically, $\Xi_{c}$ and $\Xi^{\prime}_{c}$ were identified as SU(3)F
antitriple and sextet of flavor symmetry, respectively. For these kinds of
heavy baryons, usually the two light quarks $q$ (denotes $u$ or $d$) and $s$
are assumed to form a subsystem called diquark Ebert:2006rp ; Korner:1992wi .
In previous literature, a presupposition is the spin-flavor structure of
$\Xi_{c}$ is $[qs]_{0}c$ and that of $\Xi^{{}^{\prime}}_{c}$ is $[qs]_{1}c$
where the subscript 0 or 1 represents the total spin of the $qs$ subsystem.

However, recent experimental data such as the ratio of the branching fraction
$\Xi_{cc}\to\Xi^{\prime}_{c}\pi^{+}$ relative to that of the decay
$\Xi_{cc}\to\Xi_{c}\pi^{+}$ LHCb:2022rpd  indicate the spin-flavor structures
of $\Xi_{c}$ and $\Xi^{\prime}_{c}$ are not pure $[qs]_{0}c$ and $[qs]_{1}c$
(they are denoted as $\Xi^{\bar{3}}_{c}$ and $\Xi^{6}_{c}$, respectively,
where the superscripts $\bar{3}$ and $6$ correspond to SU(3)F anti-triple and
sextet) but mixtures of them. In this case, we can use a mixing angle $\theta$
($0<\theta<\pi$) to describe the composition of $\Xi^{\bar{3}}_{c}$ and
$\Xi^{6}_{c}$ in flavor-spin wave functions of $\Xi_{c}$ and
$\Xi^{\prime}_{c}$, i.e., $|\Xi_{c}\rangle={\rm
cos}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm sin}\theta\,|\Xi^{6}_{c}\rangle$
and $|\Xi^{\prime}_{c}\rangle=-{\rm sin}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm
cos}\theta\,|\Xi^{6}_{c}\rangle$. When $\theta$ is set to 0, i.e.,
$\Xi_{c}=\Xi^{\bar{3}}_{c}$ and $\Xi^{{}^{\prime}}_{c}=\Xi^{6}_{c}$, the
structures of $\Xi_{c}$ and $\Xi^{{}^{\prime}}_{c}$ restore the original
setting supposed by the authors of Ref. Korner:1992wi ; Ebert:2006rp . Such
mixing is a reflection of the breaking of SU(3) flavor symmetry, which is due
to the mass difference between $s$ and $u,d$ quarks.

To determine the mixing angle between $\Xi_{c}$ and $\Xi^{{}^{\prime}}_{c}$,
different approaches has been proposed while a consensus has not been
achieved. One approach is to analyze transition rates of these state based on
phenomenological models. In Ref. Ke:2022gxm , we fixed the mixing angle to be
$16.27^{\circ}\pm 2.30^{\circ}$ or $85.54^{\circ}\pm 2.30^{\circ}$ by fitting
the data
$\Gamma(\Xi_{cc}\to\Xi_{c}^{{}^{\prime}+}\pi^{+})/\Gamma(\Xi_{cc}\to\Xi_{c}\pi^{+})$.
Geng et. al. Geng:2022yxb ; Geng:2022xfz  obtain the value is $24.66^{\circ}$
from the mass spectra. Another approach is lattice simulation. Recent lattice
calculation Liu:2023feb  prefers $1.2^{\circ}\pm 0.1^{\circ}$, which is much
smaller than those from phenomenological analysis.

To better understand the mixing mechanism and obtain a precise measurement of
mixing angle of $\Xi_{c}$ and $\Xi^{{}^{\prime}}_{c}$, more experimental data
involving $\Xi_{c}$ and $\Xi^{\prime}_{c}$ are needed. Inspired by recent data
from LHCb on $\Xi_{b}\to\Xi_{c}D_{s}$, here we apply light-front quark model
Jaus ; Ji:1992yf ; Cheng:1996if ; Cheng:2003sm ; Lu:2007sg ; Li:2010bb ;
Ke:2009ed ; Ke:2010 ; Choi:2007se ; Ke:2009mn ; Ke:2011fj ; Ke:2011mu ;
Ke:2011jf ; Chua:2018lfa ; Yu:2017zst ; Ke:2017eqo ; pentaquark1 ; pentaquark2
; Tawfiq:1998nk ; Chang:2018zjq ; Geng:2022xpn  framework to study the
semileptonic and non-leptonic decays of $\Xi_{b}\to\Xi_{c}$ and
$\Xi_{b}\to\Xi^{\prime}_{c}$, and estimate the feasibility to extract the
mixing angle from these channels.

The light-front quark model has been widely used to study light-quark systems.
In our earlier work we extended the light-quark model to study baryon with
heavy quark-diquark picture Ke:2007tg ; Ke:2012wa ; Wei:2009np . Later the
three-quark picture of baryon was adopted Ke:2019smy ; Ke:2019lcf ; Lu:2023rmq
. In this study we also apply the three-quark picture to analyze decays of
heavy-baryon. In the calculation, we apply the spectator approximation to the
light diquark components.

This paper is organized as follows: after the introduction, in section II we
discuss the form factors for the transition
$\Xi_{b}\rightarrow\Xi^{(^{\prime})}_{c}$ in the light-front quark model. Our
numerical results for $\Xi_{b}\rightarrow\Xi^{(^{\prime})}_{c}$ are presented
in section III. The section IV is devoted to our conclusion and discussions.

##  II the form factors of $\Xi_{b}\to\Xi_{c}$ and
$\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$ in LFQM

The leading Feynman diagram responsible for the weak decay
$\Xi_{b}\to\Xi_{c}^{(^{\prime})}$ is shown in Fig. 1. During this process, $b$
quark decays to $c$ quark and diquark $sq$ stands as the spectator. Here
spectator approximation is directly applied to diquark state $sq$ because its
spin configuration does not change during the transition at the leading order,
which greatly alleviates the theoretical difficulties for calculating the
hadronic transition matrix elements.

---
Figure 1: The Feynman diagrams for $\Xi_{b}\to\Xi^{(^{\prime})}_{c}$
transitions, where $Q_{1}$ and $Q_{2}$ denote heavy quark in the initial and
final states, respectively. $q_{1}$ and $q_{2}$ represent light quark states.
$\bullet$ denotes $V-A$ current vertex.

The form factors for the weak transition $\Xi_{b}\rightarrow\Xi_{c}$ are
defined as

$\displaystyle\langle\Xi_{c}(P^{\prime},S^{\prime},S_{z}^{\prime})\mid\bar{Q^{\prime}}\gamma_{\mu}(1-\gamma_{5})Q\mid\Xi_{b}(P,S,S_{z})\rangle$
(1) $\displaystyle=$
$\displaystyle\bar{u}_{\Xi_{c}}(P^{\prime},S^{\prime}_{z})\left[\gamma_{\mu}f_{1}(q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{\Xi_{b}}}f_{2}(q^{2})+\frac{q_{\mu}}{M_{\Xi_{b}}}f_{3}(q^{2})\right]u_{\Xi_{b}}(P,S_{z})$
$\displaystyle-\bar{u}_{\Xi_{c}}(P^{\prime},S^{\prime}_{z})\left[\gamma_{\mu}g_{1}(q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{\Xi_{b}}}g_{2}(q^{2})+\frac{q_{\mu}}{M_{\Xi_{b}}}g_{3}(q^{2})\right]\gamma_{5}u_{\Xi_{b}}(P,S_{z}).$

where $P$ and $P^{\prime}$ denote four-momentum of $\Xi_{b}$ and $\Xi_{c}$,
respectively. $q\equiv P-P^{\prime}$.

In terms of the spin-flavor structures of $\Xi_{b}$ and $\Xi_{c}$, we assumed
$|\Xi_{c}\rangle$ is a mixture of state $|\Xi^{\bar{3}}_{c}\rangle$ and
$|\Xi^{6}_{c}\rangle$, with a mixing angle $\theta$ Ke:2022gxm :

$|\Xi_{c}\rangle={\rm cos}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm
sin}\theta\,|\Xi^{6}_{c}\rangle\ ,$ (2)

and for $|\Xi^{{}^{\prime}}_{c}\rangle$:

$|\Xi^{{}^{\prime}}_{c}\rangle=-{\rm
sin}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm cos}\theta\,|\Xi^{6}_{c}\rangle\ .$
(3)

With heavy-quark symmetry, we can also express $|\Xi_{b}\rangle$ as a
superposition of flavor eigenstates:

$|\Xi_{b}\rangle={\rm cos}\theta\,|\Xi^{\bar{3}}_{b}\rangle+{\rm
sin}\theta\,|\Xi^{6}_{b}\rangle\ ,$ (4)

Here we take the limit of heavy-quark symmetry, hence the mixing angle
$\theta$ is the same for $b$-quark baryons.

Using eigenstates of $\text{SU(3)}_{F}$, $|\Xi^{6}_{b}\rangle$,
$|\Xi^{\bar{3}}_{b}\rangle$, $|\Xi^{\bar{3}}_{c}\rangle$,
$|\Xi^{6}_{c}\rangle$, the matrix element
$\langle\Xi_{c}(P^{\prime},S^{\prime},S_{z}^{\prime})\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi_{b}(P,S,S_{z})\rangle$
can be written as

${\rm
cos}^{2}\theta\langle\Xi^{\bar{3}}_{c}\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi^{\bar{3}}_{b}\rangle+{\rm
sin}^{2}\theta\langle\Xi^{6}_{c}\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi^{6}_{b}\rangle.$
(5)

where we neglect the transition matrix elements for
$\Xi^{\bar{3}}_{b}\to\Xi^{6}_{c}$ and $\Xi^{6}_{b}\to\Xi^{\bar{3}}_{c}$
because they are forbidden in the leading-order approximation of light-front
quark model. In certain phenomenological approaches the transitions
$\Xi^{\bar{3}}_{b}\to\Xi^{6}_{c}$ and $\Xi^{6}_{b}\to\Xi^{\bar{3}}_{c}$ maybe
exist in higher order of $\alpha_{s}$ but they should be very suppressed like
the decays of $\Lambda_{b}\to\Sigma_{c}$.

For the transition matrix elements
$\langle\Xi^{\bar{3}}_{c}\mid\bar{s}\gamma_{\mu}(1-\gamma_{5})c\mid\Xi^{\bar{3}}_{b}\rangle$,
the corresponding form factors are denoted as $f_{i}^{s}$, $g_{i}^{s}$ (See
Eq. (1) for the definition of form factors). Similarly, for
$\langle\Xi^{6}_{c}\mid\bar{s}\gamma_{\mu}(1-\gamma_{5})c\mid\Xi^{6}_{b}\rangle$,
the form factors are denoted as $f_{i}^{v}$ , $g_{i}^{v}$. Combining Eq. (1)
and Eq. (5), we have

$\displaystyle f_{i}={\rm cos}^{2}\theta f^{s}_{i}+{\rm sin}^{2}\theta
f^{v}_{i},$ $\displaystyle g_{i}={\rm cos}^{2}\theta g^{s}_{i}+{\rm
sin}^{2}\theta g^{v}_{i}.$ (6)

Following the procedures given in Refs. pentaquark1 ; pentaquark2 ; Ke:2007tg
; Ke:2012wa , the transition matrix element can be computed using light-front
approach with specific forms of vertex functions of $\Xi^{\bar{3}}_{b}$,
$\Xi^{\bar{3}}_{c}$, $\Xi^{6}_{b}$ and $\Xi^{6}_{c}$. The calculations for
$\Xi^{\bar{3}}_{b}\to\Xi^{\bar{3}}_{c}$ and $\Xi^{6}_{b}\to\Xi^{6}_{c}$ are
same as those for $\Lambda_{b}\to\Lambda_{c}$ and $\Sigma_{b}\to\Sigma_{c}$,
respectively. The detailed derivation and the expressions $f_{i}^{s}$,
$g_{i}^{s}$, $f_{i}^{v}$, $g_{i}^{v}$ can be found in our earlier paper
Ke:2019smy  .

For the transition matrix element
$\langle\Xi^{{}^{\prime}}_{c}(P^{\prime},S^{\prime},S_{z}^{\prime})\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi_{b}(P,S,S_{z})\rangle$,
by using $|\Xi^{6}_{b}\rangle$, $|\Xi^{\bar{3}}_{b}\rangle$,
$|\Xi^{\bar{3}}_{c}\rangle$, $|\Xi^{6}_{c}\rangle$, it can be written as

$-{\rm cos}\theta{\rm
sin}\theta\langle\Xi^{\bar{3}}_{c}\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi^{\bar{3}}_{b}\rangle+{\rm
cos}\theta{\rm
sin}\theta\langle\Xi^{6}_{c}\mid\bar{c}\gamma_{\mu}(1-\gamma_{5})b\mid\Xi^{6}_{b}\rangle.$
(7)

The form factors of transition matrix element are also defined as in Eq. (1).
Here we just add a symbol “ ′ ” on $f_{1}$, $f_{2}$, $g_{1}$ and $g_{2}$ to
distinguish the quantities for $\Xi_{b}\to\Xi^{\prime}_{c}$ and those for
$\Xi_{b}\to\Xi_{c}$. They are given by

$\displaystyle f^{\prime}_{i}={\rm cos}\theta{\rm
sin}\,\theta(f^{v}_{i}-f^{s}_{i}),$ $\displaystyle g^{\prime}_{i}={\rm
cos}\theta{\rm sin}\,\theta(g^{v}_{i}-g^{s}_{i}).$ (8)

##  III Numerical Results

###  III.1 The results for $\Lambda_{b}\to\Lambda_{c}$

In Ref. Ke:2019smy  we extended three-quark picture to study the decay of
$\Lambda_{b}\to\Lambda_{c}$ and $\Sigma_{b}\to\Sigma_{c}$. In that paper we
employ a pole form in Eq. (9) to fix the form factor of the transition.

$\displaystyle
F(q^{2})=\frac{F(0)}{\left(1-\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)\left[1-a\left(\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)+b\left(\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)^{2}\right]},$
(9)

where $M_{\mathcal{B}_{i}}$ is the mass of the initial baryon.

Later we find it doesn’t work for the decay of double charmed baryon so in
Refs. Ke:2019lcf ; Lu:2023rmq  we employed a polynomial to parameterize these
form factors $f^{s}_{i}$, $g^{s}_{i}$, $f^{v}_{i}$ and $g^{v}_{i}$ ($i=1,2$),

$\displaystyle
F(q^{2})=F(0)\left[1+a\left(\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)+b\left(\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)^{2}\right].$
(10)

In this paper we first check whether the above polynomial formula also can be
used for the transition $\Lambda_{b}\to\Lambda_{c}$. With the same parameters
(quark masses and $\beta$ parameters in wave functions) in Ref. Ke:2019smy  we
refit the parameters in Eq. (10). The fitted values of $a,~{}b$ and $F(0)$ in
the form factors are presented in Table 1. We also depict two results in Fig.
2. The left graph (a) is the result from Ref.Ke:2019smy  and the right one (b)
is the result this time. Comparing these plots, one can find they are nearly
equal to each other. With these form factors, we re-calculate the decay rates
$\Gamma$, the integrated longitudinal and transverse asymmetries $a_{l}$ and
$a_{t}$ and their ratio $R=a_{l}/a_{t}$ of semi-leptonic for the transtion
$\Lambda_{b}\to\Lambda_{c}$ and the results are present in table. 2. We also
collect the results in Refs. Ke:2007tg ; Ke:2019smy  for comparison. From the
values in Tab. 2, we can conclude that two extrapolative forms of the form
factors don’t significantly affect the results of transitions of
$\Lambda_{b}\to\Lambda_{c}$. We also reevaluate the width and up-down
asymmetry $\alpha$ of several no-leptonic decay channels. The results of the
nonleptonic decays in table 3 are also very consistent. For the detailed
expressions used in the calculation of these physical quantities, one can
refer to Refs. Cheng:1996cs ; Korner:1994nh ; Korner:1991ph ; Bialas:1992ny
and appendix of the Ref. Ke:2019smy .

Figure 2: Form factors for the decay $\Lambda_{c}\to\Lambda_{c}$ (a) pole form and (b) polynomial form. Table 1: The form factors given in polynomial form for the transition $\Lambda_{b}\to\Lambda_{c}$. $F$ | $F(0)$ | $a$ | $b$
---|---|---|---
$f^{s}_{1}$ | 0.488 | 1.94 | 1.70
$f^{s}_{2}$ | -0.181 | 2.44 | 2.52
$g^{s}_{1}$ | 0.470 | 1.86 | 1.58
$g^{s}_{2}$ | -0.048 | 2.75 | 3.07
Table 2: The widths (in unit of $10^{10}{\rm s}^{-1}$) and polarization asymmetries of $\Lambda_{b}\to\Lambda_{c}l\bar{\nu}_{l}$ . | $\Gamma$ | $a_{L}$ | $a_{T}$ | $R$ | $P_{L}$
---|---|---|---|---|---
The results in Ke:2019smy  | 4.22 | -0.962 | -0.766 | 1.54 | -0.885
The results in Ke:2007tg  | 5.15 | -0.932 | -0.601 | 1.47 | -0.798
This work | 4.36 | -0.933 | -0.668 | 1.49 | -0.826
Table 3: The widths (in unit of $10^{10}{\rm s}^{-1}$) and up-down asymmetries of non-leptonic decays $\Lambda_{b}\to\Lambda_{c}M$. | the results in Ref. Ke:2019smy  | the results in Ke:2007tg  | this work
---|---|---|---
| $\Gamma$ | $\alpha$ | $\Gamma$ | $\alpha$ | $\Gamma$ | $\alpha$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\pi^{-}$ | $0.261$ | $-0.999$ | $0.307$ | $-1$ | $0.261$ | $-0.999$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\rho^{-}$ | $0.769$ | $-0.875$ | $0.848$ | $-0.883$ | $0.774$ | $-0.875$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}K^{-}$ | $0.0209$ | $-0.999$ | $0.0247$ | $-1$ | $0.0210$ | $-0.999$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}K^{*-}$ | $0.0398$ | $-0.836$ | $0.0440$ | $-0.846$ | $0.0402$ | $-0.837$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}a_{1}^{-}$ | $0.758$ | $-0.710$ | $0.838$ | $-0.726$ | $0.770$ | $-0.710$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}D_{s}^{-}$ | $0.927$ | $-0.974$ | $0.932$ | $-0.982$ | $0.928$ | $-0.987$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}D^{*-}_{s}$ | $1.403$ | $-0.327$ | $1.566$ | $-0.360$ | $1.446$ | $-0.330$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}D^{-}$ | $0.0355$ | $-0.979$ | $0.0410$ | $-0.986$ | $0.0357$ | $-0.989$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}{D^{*}}^{-}$ | $0.0630$ | $-0.371$ | $0.0702$ | $-0.403$ | $0.0649$ | $-0.374$

###  III.2 The results for $\Xi_{b}\to\Xi_{c}$ and
$\Xi_{b}\to\Xi^{{}^{\prime}}_{c}$

After checking the effectiveness of the polynomial we calculate the form
factors for $\Xi^{\bar{3}}_{b}\to\Xi^{\bar{3}}_{c}$ and
$\Xi^{6}_{b}\to\Xi^{6}_{c}$ in space-like region and extrapolate them to time-
like region by the Eq. (10). The parameters in the expressions of the form
factors are same as those for $\Lambda_{b}\to\Lambda_{c}$ in Ref. Ke:2019smy
except $\beta_{23}\approx 2.9\beta_{us}=0.994$ GeV. We also depict them in
Fig. 3. One can find the form factors for the transition
$\Xi^{\bar{3}}_{b}\to\Xi^{\bar{3}}_{c}$ are close to those for
$\Lambda_{b}\to\Lambda_{c}$.

Figure 3: Form factors for the decay $\Xi^{\bar{3}}_{b}\to\Xi^{\bar{3}}_{c}$(a) and $\Xi^{6}_{b}\to\Xi^{6}_{c}$ (b). Table 4: The form factors given in polynomial form for the transitions $\Xi^{\bar{3}}_{b}\to\Xi^{\bar{3}}_{c}$ and $\Xi^{6}_{b}\to\Xi^{6}_{c}$ . $F$ | $F(0)$ | $a$ | $b$
---|---|---|---
$f^{s}_{1}$ | 0.467 | 2.19 | 2.09
$f^{s}_{2}$ | -0.185 | 2.70 | 3.01
$g^{s}_{1}$ | 0.448 | 2.09 | 1.92
$g^{s}_{2}$ | -0.052 | 3.32 | 4.54
$f^{v}_{1}$ | 0.471 | 2.57 | 2.40
$f^{v}_{2}$ | 0.378 | 2.56 | 2.81
$g^{v}_{1}$ | -0.149 | 2.07 | 2.00
$g^{v}_{2}$ | -0.006 | 2.95 | 2.98
Table 5: The widths (in unit of $10^{10}{\rm s}^{-1}$) and polarization asymmetries of $\Xi_{b}\to\Xi_{c}l\bar{\nu}_{l}$ . | $\Gamma$ | $a_{L}$ | $a_{T}$ | $R$ | $P_{L}$
---|---|---|---|---|---
$\theta=0^{\circ}$ | 3.68 (0.844) | -0.962 (1.03) | -0.745(1.12) | 1.63(1.09) | -0.879 (1.06)
$\theta=16.27^{\circ}$ | 3.13 (0.718) | -0.973 (1.04) | -0.756(1.13) | 1.78(1.19) | -0.895(1.08)
$\theta=85.54^{\circ}$ | 1.15 (0.264) | 0.685(-0.734) | -0.237(0.355) | 6.69(4.49) | 0.565(-0.684)

Figure 4:  Differential decay rates $d\Gamma/d\omega$ for the decay
$\Xi_{b}\to\Xi_{c}l\bar{\nu}_{l}$ (a) and
$\Xi_{b}\to\Xi^{{}^{\prime}}_{c}l\bar{\nu}_{l}$ (b).

Pre-setting different mixing angles, we evaluate the rates and some
asymmetries parameters for the semilepotonic decay and nonleptonic decays of
$\Xi_{b}\to\Xi_{c}$. In tables 5 and 6 we explicitly list the theoretical
predictions with different mixing angles. The dependence of differential decay
widths $d\Gamma/d\omega$ ($\omega=\frac{P\cdot P^{\prime}}{mm^{\prime}}$) on
$\omega$ are depicted in Fig. 4 (a). In order to minimize the model dependence
of results, we also calculate the ratio of these values to those for
$\Lambda_{b}\to\Lambda_{c}l\bar{\nu}_{l}$ which are presented in the
parentheses in tables 5333In table 6, 7 and 8 the value in every parenthes is
the ratio with respect to that of $\Lambda_{b}\to\Lambda_{c}$. One may notice
when $\theta=0^{\circ}$ and $\theta=16.27^{\circ}$ the polarization
asymmetries of $\Xi_{b}\to\Xi_{c}l\bar{\nu}_{l}$ are very close to each other
and the decay widths has a slight difference. For a large mixing angle such as
$\theta=85.54^{\circ}$, the mixing angle apparently changes the decay rate and
polarization asymmetries relative to the case at $\theta=0^{\circ}$ in the
decay $\Xi_{b}\to\Xi_{c}l\bar{\nu}_{l}$. In terms of
$\mathcal{R}(\frac{\Xi^{0}_{b}}{\Lambda_{b}})$, we find the ratio
$\frac{\sigma(\Xi_{b}^{0})}{\sigma(\Lambda^{0}_{b})}\approx 18.4$ when
$\theta$ takes $16.27^{\circ}$.

For the transition $\Xi_{b}\to\Xi^{\prime}_{c}$, when $\theta=0^{\circ}$, the
theoretical results are zero in our calculation since the spin of spectator
doesn’t change. Certainly there maybe exist a very small decay rate for the
transition of spin-flip in some other approaches but the situation should be
similar to the case of the transition $\Lambda_{b}\to\Sigma_{c}$ which is very
small so that it isn’t listed in PDG databook ParticleDataGroup:2022pth .

If the mixing angle $\theta=16.27^{\circ}$ the semileptonic decay of
$\Xi_{b}\to\Xi^{\prime}_{c}$ will be one order smaller than that of
$\Xi_{b}\to\Xi_{c}$ and the nonleptonic decay of $\Xi_{b}\to\Xi^{\prime}_{c}$
will fifteen to twenty times smaller than that of $\Xi_{b}\to\Xi_{c}$. The
dependence of the differential decay widths $d\Gamma/d\omega$
($\omega=\frac{P\cdot P^{\prime}}{mm^{\prime}}$) on $\omega$ are depicted in
Fig. 4 (b). Comparing Fig. 4 (a) and (b) one also find the ratio of the
differential decay widths at every $\omega$ is also about 10. In Ref.
LHCb:2023ngz  the signal yields for $\Xi^{0}_{b}$ and $\Xi^{+}_{b}$ decay are
$462\pm 29$ and $175\pm 14$, respectively. It seems that we have opportunity
to observe the decays such as $\Xi_{b}\to\Xi^{\prime}_{c}l\nu_{l}$,
$\Xi_{b}\to\Xi^{\prime}_{c}D_{s}$, $\Xi_{b}\to\Xi^{\prime}_{c}D^{*}_{s}$ and
(or) $\Xi_{b}\to\Xi^{\prime}_{c}\rho$ in the near future. These processes are
very optimal to determine the mixing angle $\theta$ since
$\Xi_{b}\to\Xi^{\prime}_{c}$ is very small when $\theta=0^{\circ}$. Once an
observation $\Xi_{b}\to\Xi^{\prime}_{c}$ is confirmed, we will have strong
evidence to believe that a modest value of mixing angle $\theta$.

In Tables 7 and 8 we only list the results at $\theta=16.27^{\circ}$. For
other mixing angles the decay rates are different but the polarization
asymmetries are same. The results can be understood from the expressions of
the form factors $f^{\prime}_{i}$ and $g^{\prime}_{i}$ in Eq. (II). To
calculate the decay rate of $\Xi_{b}\to\Xi^{\prime}_{c}$ for other value of
$\theta$, one can simply re-scale the width in tables 7 or 8 by a factor ${\rm
sin\theta}{\rm cos\theta}/({\rm sin16.27^{\circ}}{\rm cos16.27^{\circ}}$).

Table 6: The widths (in unit $10^{10}{\rm s}^{-1}$) and up-down asymmetries of non-leptonic decays $\Xi_{b}\to\Xi_{c}M$. | $\theta=0^{\circ}$ | $\theta=16.27^{\circ}$ | $\theta=85.54^{\circ}$
---|---|---|---
| $\Gamma$ | $\alpha$ | $\Gamma$ | $\alpha$ | $\Gamma$ | $\alpha$
$\Xi_{b}^{0}\to\Xi_{c}\pi^{-}$ |  $0.247$ (0.946) |  $-0.999$ (1.000) |  $0.224$ (0.858) |  $-0.988$ (0.989) |  $0.143$ (0.548) |  $0.557$ (-0.558)
$\Xi_{b}^{0}\to\Xi_{c}\rho^{-}$ |  $0.725$ (0.937) |  $-0.877$ (1.002) |  $0.651$ (0.841) |  $-0.880$ (1.006) |  $0.388$ (0.501) |  $0.567$ (-0.648)
$\Xi_{b}^{0}\to\Xi_{c}K^{-}$ |  $0.0198$ (0.943) |  $-0.999$ (1.000) |  $0.0179$ (0.852) |  $-0.987$ (0.988) |  $0.0116$ (0.552) |  $0.551$ (-0.552)
$\Xi_{b}^{0}\to\Xi_{c}K^{*-}$ |  $0.0375$ (0.933) |  $-0.839$ (1.002) |  $0.0336$ (0.836) |  $-0.846$ (1.011) |  $0.0195$ (0.485) |  $0.571$ (-0.682)
$\Xi_{b}^{0}\to\Xi_{c}a_{1}^{-}$ |  $0.712$ (0.925) |  $-0.715$ (1.007) |  $0.641$ (0.832) |  $-0.733$ (1.032) |  $0.338$ (0.439) |  $0.583$ (-0.821)
$\Xi_{b}^{0}\to\Xi_{c}D_{s}^{-}$ |  $0.857$ (0.923) |  $-0.972$ (0.986) |  $0.796$ (0.858) |  $-0.941$ (0.954) |  $0.612$ (0.659) |  $0.455$ (-0.461)
$\Xi_{b}^{0}\to\Xi_{c}D^{*-}_{s}$ |  $1.271$ (0.879) |  $-0.338$ (1.024) |  $1.090$ (0.754) |  $-0.377$ (1.142) |  $0.418$ (0.289) |  $0.643$ (1.948)
$\Xi_{b}^{0}\to\Xi_{c}D^{-}$ |  $0.0330$ (0.924) |  $-0.977$ (0.988) |  $0.0305$ (0.854) |  $-0.948$ (0.959) |  $0.0230$ (0.644) |  $0.466$ (-0.471)
$\Xi_{b}^{0}\to\Xi_{c}{D^{*}}^{-}$ |  $0.0575$ (0.886) |  $-0.382$ (1.021) |  $0.0495$ (0.763) |  $-0.420$ (1.122) |  $0.0199$ (0.183) |  $0.636$ (1.701)
Table 7: The widths (in unit of $10^{10}{\rm s}^{-1}$) and polarization asymmetries of $\Xi_{b}\to\Xi^{\prime}_{c}l\bar{\nu}_{l}$ . | $\Gamma$ | $a_{L}$ | $a_{T}$ | $R$ | $P_{L}$
---|---|---|---|---|---
$\theta=16.27^{\circ}$ | 0.300(0.0668) | -0.0729(0.0781) | -0.622(0.931) | 0.997 (0.669) | -0.348(0.421)
Table 8: The widths (in unit $10^{10}{\rm s}^{-1}$) and up-down asymmetries of non-leptonic decays $\Xi_{b}\to\Xi^{\prime}_{c}M$ with $\theta=16.27^{\circ}$. | $\Gamma$ | $\alpha$
---|---|---
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}\pi^{-}$ |  $0.0142$ (0.0544) |  $0.0138$ (-0.0138)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}\rho^{-}$ |  $0.0450$ (0.0581) |  $0.100$ (-0.114)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}K^{-}$ |  $0.00112$ (0.0533) |  $0.0180$ (-0.0180)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}K^{*-}$ |  $0.00238$ (0.0592) |  $0.124$ (-0.148)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}a_{1}^{-}$ |  $0.0487$ (0.0632) |  $0.194$ (-0.273)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}D_{s}^{-}$ |  $0.0373$ (0.0402) |  $0.0902$ (-0.0914)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}D^{*-}_{s}$ |  $0.107$ (0.0740) |  $0.325$ (-0.985)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}D^{-}$ |  $0.00149$ (0.0417) |  $0.0818$ (-0.0827)
$\Xi_{b}^{0}\to\Xi^{\prime}_{c}{D^{*}}^{-}$ |  $0.00474$ (0.0730) |  $0.317$ (-0.848) z

##  IV Conclusions and discussions

In this work we study the transition rates of
$\Xi_{b}\to\Xi^{(^{\prime})}_{c}$ in the light front quark model with a three-
quark picture of baryon. To correctly compute the transition rates, it is
important to properly assess the spin-flavor structures of initial and final
states. In the three-quark picture, the two light quarks constitute a diquark
which combine with a heavy quark to form baryons $\Xi_{b}$ and $\Xi_{c}$. In
previous studies, the light $qs$ pair in $\Xi_{c}$ ($\Xi^{{}^{\prime}}_{c}$ )
was are assumed be to the eigenstates (scalar or pseudovector ) of SU(3)
flavor symmetry. However, experimental measurements such as the ratio
$\Gamma(\Xi^{++}_{cc}\to\Xi^{+}_{c}\pi)/\Gamma(\Xi^{++}_{cc}\to\Xi^{{}^{\prime}+}_{c}\pi)$
LHCb:2022rpd  suggest the spin-flavor structures of $\Xi_{c}$ and
$\Xi^{\prime}_{c}$ should be a mixture of $\Xi^{\bar{3}}_{c}$ and
$\Xi^{6}_{c}$. Therefore, here we introduce a mixing angle $\theta$ to define
the state $|\Xi_{c}\rangle$ as ${\rm
cos}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm sin}\theta\,|\Xi^{6}_{c}\rangle$,
and $|\Xi^{{}^{\prime}}_{c}\rangle$ as $-{\rm
sin}\theta\,|\Xi^{\bar{3}}_{c}\rangle+{\rm cos}\theta\,|\Xi^{6}_{c}\rangle$.
Because of heavy quark symmetry, we can similarly define $|\Xi_{b}\rangle$ as
a mixture of states with the same mixing angle: ${\rm
cos}\theta\,|\Xi^{\bar{3}}_{b}\rangle+{\rm sin}\theta\,|\Xi^{6}_{b}\rangle$.

Based on proposed spin-flavor structures of $\Xi_{b}$, $\Xi_{c}$ and
$\Xi^{{}^{\prime}}_{c}$, we calculate semileptonic and nonleptonic decay of
$\Xi_{b}\to\Xi_{c}$ with specific value of mixing angle ($\theta$ =
$0^{\circ}$, $16.27^{\circ}$ and $85.54^{\circ}$) used in previous analysis
Ke:2022gxm . We find the decay width and the asymmetry parameter at
$\theta=85.54^{\circ}$ are very different with those without mixing
($\theta=0^{\circ}$). However for the mixing angle $\theta=16.27^{\circ}$,
only the decay width has a significant decrease relative to the result without
mixing.

Following this theoretical framework, we also evaluate the rates of
semileptonic decay and nonleptonic decay of $\Xi_{b}\to\Xi^{\prime}_{c}$.
Since the form factors $f^{\prime}_{i}={\rm cos}\theta{\rm
sin}\,\theta(f^{v}_{i}-f^{s}_{i})$, $g^{\prime}_{i}={\rm cos}\theta{\rm
sin}\,\theta(g^{v}_{i}-g^{s}_{i})$, the theoretical decay rates are
proportional to the value ${\rm cos}^{2}\theta{\rm sin}^{2}\,\theta$, hence we
only display results with $\theta=16.27^{\circ}$ for the purpose of
illustration. Our numerical results indicate that the decay rates of these
decays are about ten to twenty times smaller than the values of the
transitions $\Xi_{b}\to\Xi_{c}$, which can be feasible to measure
experimentally in the near future. Without a mixing of flavor eigenstates, the
rates of the transitions $\Xi_{b}\to\Xi^{\prime}_{c}$ should be very tiny
(about the same order of the transition of $\Lambda_{b}\to\Sigma_{c}$).
However, a modest mixing angle such as $\theta=16.27^{\circ}$ will make the
transition rate large enough for measurements, which makes these channels
ideal for studying the mixing of $\Xi_{c}$ and $\Xi^{\prime}_{c}$ ($\Xi_{b}$
and $\Xi^{\prime}_{b}$). We hope measurement of those decay channels of
$\Xi_{b}\to\Xi^{\prime}_{c}$ will be available in the near future, which will
help us to test flavor mixing angle $\theta$ and elucidate the mechanism of
decay of heavy baryons.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
(NNSFC) under the contract No. 12075167.

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Compact homogeneous locally conformally Kähler manifolds are Vaisman. A new
proof.
Liviu Ornea111Liviu Ornea is partially supported by Romanian Ministry of
Education and Research, Program PN-III, Project number PN-III-P4-ID-
PCE-2020-0025, Contract 30/04.02.2021, Misha Verbitsky222Misha Verbitsky is
partially supported by by the HSE University Basic Research Program, FAPERJ
E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Keywords: Locally conformally Kähler, homogeneous, LCK potential, Vaisman
manifold, holomorphic action. 2010 Mathematics Subject Classification: 53C55,
53C30.

Abstract

An LCK manifold with potential is a complex manifold with a Kähler potential
on its cover, such that any deck transformation multiplies the Kähler
potential by a constant multiplier. We prove that any homogeneous LCK manifold
admits a metric with LCK potential. This is used to give a new proof that any
compact homogeneous LCK manifold is Vaisman.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Locally conformally Kähler manifolds
    2. 2.2 Vaisman manifolds
    3. 2.3 LCK manifolds with potential
  3. 3 Homogeneous LCK manifolds admit LCK potential
  4. 4 Homogeneous Vaisman manifolds

##  1 Introduction

A locally conformally Kähler (LCK) manifold is a complex manifold $(M,I)$
equipped with a Hermitian metric $g$ (LCK metric) which is locally conformally
equivalent to a Kähler metric. Then the Hermitian form $\omega(x,y):=g(Ix,y)$
satisfies $d\omega=\theta\wedge\omega$, where $\theta$ is a 1-form, called the
Lee form (see [DO] for an introduction to the subject).

An LCK metric $g$ is called Vaisman if the Lee form is not only closed, but
also parallel with respect to the Levi Civita connection of $g$.

The LCK condition is conformally invariant: if $g$ is an LCK metric on $M$,
then $e^{f}g$ is LCK for all smooth functions $f$ on $M$. By contrast, the
Vaisman condition is not conformally invariant. Moreover, on a compact complex
manifold, a Vaisman metric (if it exists) is unique up to homothety in its
conformal class and coincides with the Gauduchon metric of that conformal
class.

One of the most interesting problems in LCK geometry is to find sufficient
conditions for an LCK manifold to admit Vaisman metrics. In this paper we
prove that homogeneity is such a sufficient condition.

An LCK manifold $(M,I,g)$ is homogeneous if it admits a transitive and
effective action of a Lie group by holomorphic isometries of the LCK metric.
In this case, $M=G/H$, where $G$ is a connected Lie group, and $H$ is the
stabilizer subgroup of $G$.

Remark 1.1: (i) A group $G$ as above also preserves $\omega$ and $\theta$.

(ii) Recall that the group $\operatorname{Aut}(M)$ of biholomorphic
conformalities of a compact LCK manifold is compact. Indeed, any holomorphic
conformality preserves the corresponding Gauduchon metric which is unique up
to a constant. We can therefore assume that $M$ is homogeneous under the group
of holomorphic conformalities.

The following result was announced in [HK1], see also [HK2], [HK3] and [Gu]
for a subsequent discussion. A short, complete and self-contained proof
appeared in [GMO]. Here we present a new proof, based on the notion of LCK
structure with potential.

Theorem 1.2:  A compact homogeneous locally conformally Kähler manifold is
Vaisman.

The new proof amounts in showing that the homogeneity implies the existence of
a holomorphic circle action whose lift to the Kähler cover does not contain
only homotheties of the Kähler metric. This will imply that the manifold is
LCK with potential. We then observe that the Lee form of this LCK structure
with potential has constant length, which characterizes the Vaisman metrics
among the LCK metrics with potential.

Section 2 of this paper gathers the necessary background on LCK geometry,
inlcuding Vaisman and LCK with potential metrics. In section 3 we present the
proof of 1. In the last section, we give new proofs for two classical results
about homogeneous Vaisman manifolds.

##  2 Preliminaries

###  2.1 Locally conformally Kähler manifolds

Definition 2.1: Let $(M,I)$ be a complex manifold,
$\dim_{\mathbb{C}}M\geqslant 2$. It is called locally conformally Kähler (LCK)
if it admits a Hermitian metric $g$ whose fundamental 2-form
$\omega(\cdot,\cdot):=g(\cdot,I\cdot)$ satisfies

$d\omega=\theta\wedge\omega,\quad d\theta=0,$ (2.1)

for a certain closed 1-form $\theta$ called the Lee form.

Remark 2.2: Definition (2.1) is equivalent to the existence of a covering
$\tilde{M}$ endowed with a Kähler metric $\Omega$ which is acted on by the
deck group $\operatorname{Aut}_{M}(\tilde{M})$ by homotheties.

Remark 2.3:  The operator $d_{\theta}:=d-\theta\wedge$ is called twisted de
Rham operator. It obviously satisfies $d_{\theta}^{2}=0$ and hence
$(\Lambda^{*}M,d_{\theta})$ produces a cohomology called Morse-Novikov or
twisted.

The following fundamental result of I. Vaisman shows that on compact complex
manifolds, Kähler and LCK metrics cannot coexist.

Theorem 2.4: ([Va1]) Let $(M,\omega,\theta)$ be a compact LCK manifold, not
globally conformally Kähler (i.e. with non-exact Lee form). Then $M$ does not
admit a Kähler metric.

###  2.2 Vaisman manifolds

Definition 2.5: An LCK manifold $(M,\omega,\theta)$ is called Vaisman if
$\nabla\theta=0$, where $\nabla$ is the Levi-Civita connection of $g$.

Example 2.6: All diagonal Hopf manifolds are Vaisman ([OV4]). The Vaisman
compact complex surfaces are classified in [B], see also [VVO].

There exist compact LCK manifolds which do not admit Vaisman metrics. Such are
the LCK Inoue surfaces, [B], the Oeljeklaus-Toma manifolds, [OT], [O], and the
non-diagonal Hopf manifolds, [OV4], [VVO].

Remark 2.7:  On a Vaisman manifold, the Lee field $\theta^{\sharp}$ and the
anti-Lee field $I\theta^{\sharp}$ are real holomorphic
($\operatorname{Lie}_{\theta^{\sharp}}I=\operatorname{Lie}_{I\theta^{\sharp}}I=0$)
and Killing
($\operatorname{Lie}_{\theta^{\sharp}}g=\operatorname{Lie}_{I\theta^{\sharp}}g=0$),
see [DO]. Moreover $[\theta^{\sharp},I\theta^{\sharp}]=0$. The 2-dimensional
foliation $\Sigma$ they generate is called the canonical foliation.

According to the above Remark, the Lee field generates a flow of holomorphic
isometries. The following characterization is a partial converse:

Theorem 2.8:  ([KO]) Let $(M,\omega,\theta)$ be an LCK manifold equipped with
a holomorphic and conformal ${\mathbb{C}}$-action $\rho$ without fixed points,
which lifts to non-isometric homotheties on the Kähler cover $\tilde{M}$. Then
$(M,\omega,\theta)$ is conformally equivalent with a Vaisman manifold.

Remark 2.9: Since $\theta$ is parallel, it has constant norm and thus we can
always scale the LCK metric such that $|\theta|=1.$ In this assumption, the
following formula holds, [DO]:

$d\theta^{c}=\omega-\theta\wedge\theta^{c},\quad\text{where}\quad\theta^{c}(X)=-\theta(IX).$
(2.2)

Moreover, one can see, [Ve], that the (1,1)-form $\omega_{0}:=d^{c}\theta$ is
semi-positive definite, having all eigenvalues111The eigenvalues of a
Hermitian form $\eta$ are the eigenvalues of the symmetric operator $L_{\eta}$
defined by the equation $\eta(x,Iy)=g(L_{\eta}x,y)$. positive, except one
which is 0.

###  2.3 LCK manifolds with potential

Definition 2.10: An LCK manifold has LCK potential if it admits a Kähler
covering on which the Kähler metric has a global and positive potential
function $\psi$ such that the deck group multiplies $\psi$ by a constant. In
this case, $M$ is called LCK manifold with potential.

Proposition 2.11:  The LCK manifold $(M,I,g,\theta)$ is LCK with potential if
and only if equation (2.2) is satisfied.

Definition 2.12: A function $\varphi\in C^{\infty}(M)$ is called
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic if
$\omega=d_{\theta}d^{c}_{\theta}(\varphi)$, where $d_{\theta}$ is the twisted
de Rham operator (2.1) and $d^{c}_{\theta}:=Id_{\theta}I^{-1}$.

Equation (2.2) (and hence the definition of LCK manifolds with potential) can
be translated on the LCK manifold itself:

Theorem 2.13: ([OV3, Claim 2.8]) $(M,I,\theta,\omega)$ is LCK with potential
if and only if $\omega=d_{\theta}d^{c}_{\theta}(\psi)$ for a strictly positive
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic function $\psi$ on $M$.

Remark 2.14: All Vaisman manifolds are LCK manifolds with potential: on their
Kähler covering, the automorphic potential is represented by the squared
length of the pull-back of the Lee form with respect to the Kähler metric.
Among the non-Vaisman examples, we mention the non-diagonal Hopf manifolds,
[OV3].

We shall need the following characterizations of the Vaisman metrics among the
LCK metrics with potential:

Proposition 2.15:  ([OV4]) Let $(M,\omega,\theta)$ be a compact LCK manifold
with potential. Then the LCK metric is Gauduchon if and only if
$\omega_{0}=d^{c}\theta$ is semi-positive definite, and then it is Vaisman.
Equivalently, a compact LCK manifold with potential and with constant norm of
$\theta$ is Vaisman.

In the proof that we shall present, we make use of the following sufficient
condition for a compact LCK manifold to admit an LCK metric with potential
(compare with 2.2).

Theorem 2.16:  ([OV2], [I, Theorem 0.4]) Let $M$ be a compact complex
manifold, equipped with a holomorphic $S^{1}$-action and an LCK metric $g$
(not necessarily $S^{1}$-invariant). Suppose that this $S^{1}$ action does not
lift to an isometric action on the Kähler cover of $M$. Then $g$ admits an LCK
potential.

Proof: The automorphic Kähler potential was obtained in [OV2] by cohomological
arguments, and it is not necessarily positive. In our terminology, “LCK
potential” is always positive. The existence of a positive potential is
implied a posteriori using a result from [OV5]. N. Istrati ([I, Theorem 0.4])
produced an explicit form of this argument, showing that any $S^{1}$-invariant
LCK metric, in assumptions of 2.3, admits an LCK potential.

##  3 Homogeneous LCK manifolds admit LCK potential

The idea is to show that the homogeneity of $G/H$ implies the existence of a
holomorphic circle action which is lifted to non-isometric homotheties of the
universal cover. This will imply that the manifold is LCK with potential
(2.3). We translate this potential on the manifold itself, we average it on
$G$ and obtain an invariant LCK metric with potential whose Lee form has
constant length, which characterizes the Vaisman metrics among the LCK metrics
with potential (2.3). The manifold is thus of Vaisman type. Finally, we prove
that the initial homogeneous LCK metric itself is Vaisman.

We start with the following lemma.

Lemma 3.1:  Let $(M,\omega,\theta)$ be an LCK manifold, and $A$ a vector field
acting on $M$ by holomorphic isometries. Assume that the function $\theta(A)$
is not identically zero. Then $A$ does not act by isometries when lifted to a
Kähler cover of $M$.

Proof: Let $\tilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$
be a Kähler cover of $M$. Then $\theta_{1}:=\pi^{*}\theta$ is exact:
$\theta_{1}=d\varphi$, and the corresponding Kähler form on $\tilde{M}$ can be
written as $\tilde{\omega}=e^{\varphi}\pi^{*}\omega$. Denote by $A_{1}$ the
lift of $A$ to $\tilde{M}$. Then
$\operatorname{Lie}_{A_{1}}(\tilde{\omega})=(\operatorname{Lie}_{A_{1}}\varphi)\tilde{\omega}$,
in other words, $A_{1}$ acts by isometries if and only if
$\operatorname{Lie}_{A_{1}}\varphi=0$. However, Cartan’s formula gives

$\operatorname{Lie}_{A_{1}}\varphi=A_{1}\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip
2.0ptd\varphi=d\varphi(A_{1})=\theta_{1}(A_{1}).$

Since, by assumption, $\theta(A)\neq 0$ somewhere on $M$, we have
$\theta_{1}(A_{1})\neq 0$ somewhere on $\tilde{M}$.

We can prove now that $M$ is of Vaisman type. Every homogeneous manifold $M$
can be obtained as $M=G/H$, where $G$ acts on $M$ transitively by
automorphisms, and $H$ is the stabilizer of the point. In our case, $M$ is
LCK, and $G$ acts on $M$ by holomorphic LCK isometries. Since the group of
isometries of a compact manifold is compact, we may freely assume that $G$ is
compact. This will be our running assumption from now on.

Proposition 3.2:  Let $(M,\omega,\theta)$ be a compact, homogeneous LCK
manifold, $M=G/H$, where $G$ is a compact Lie group. Then $M$ admits a
$G$-homogeneous Vaisman metric with the same Lee form.

Proof. Step 1: Let ${\mathfrak{g}}=\operatorname{Lie}(G)$. Choose
$A\in{\mathfrak{g}}$ such that $\theta(A)\neq 0$ somewhere on $M$, and let
$T\subset G$ be the closure of the one-parametric subgroup of $G$ generated by
$e^{tA}$. Then $T$ is a compact torus, hence $A$ can be obtained as a limit of
vector fields $A_{i}$ such that each one-parametric subgroup of $G$ generated
by $e^{tA_{i}}$ is a circle. Then for some $i$ the vector field $A_{i}$
satisfies $\theta(A_{1})\neq 0$ somewhere on $M$. By 3, $A_{i}$ it cannot act
by isometries on the Kähler cover of $\tilde{M}$. By 2.3, $M$ admits an LCK
metric with potential.

Step 2: By 2.3, the fundamental form of an LCK metric with potential is
written as $\omega_{\psi}=d_{\theta}d_{\theta}^{c}(\psi)$, for a positive
function $\psi$ on $M$. Averaging on the compact group $G$, we obtain a metric
$g_{1}$ with the fundamental form

$\omega_{1}=\mathrm{Av}_{G}(\omega_{\psi})=d_{\theta}d_{\theta}^{c}(\mathrm{Av}_{G}(\psi)).$

Then $g_{1}$ is still LCK with potential, but the potential is now
$G$-invariant, and the metric is also $G$-invariant. Since $\theta$ is already
$G$-invariant, the averaging process does not change it. Then $\theta$ is also
the Lee form of $g_{1}$, and its norm in this metric is constant:
$|\theta|_{\omega_{1}}=\operatorname{\text{\sf const}}$. Then 2.3 implies that
$g_{1}$ is Vaisman.

Finally, we can prove that the initial homogeneous LCK metric is itself
Vaisman.

Theorem 3.3:  Let $(M,g,I,\omega,\theta)$ be a compact, homogeneous LCK
manifold, $M=G/H$. Then $(M,\omega,\theta)$ is Vaisman.

Proof: By 3, $M$ admits a $G$-homogeneous Vaisman metric $\omega_{1}$ with the
same Lee form. By 2.3, we can assume that $\omega$ is an LCK metric with
potential. Then 3 would follow if we prove that a $G$-homogeneous metric with
potential and a given Lee form is unique up to a constant multiplier.

Proposition 3.4: Let $M=G/H$ be a $G$-homogeneous complex manifold, and
$\omega_{1},\omega_{2}$ two $G$-homogeneous LCK metrics with potential and the
same Lee form. Then $\omega_{1}$ is proportional to $\omega_{2}$.

Proof: By 2.3, $\omega_{i}=d_{\theta}d^{c}_{\theta}(\varphi_{i})$. Averaging
$\varphi_{i}$ with $G$ if necessary, we can assume that it is a $G$-invariant
function, hence constant. Then $\omega_{i}=d_{\theta}d^{c}_{\theta}(a_{i})$,
where $a_{i}$ are constant functions, hence these two forms are proportional.

##  4 Homogeneous Vaisman manifolds

For the sake of completion, we give here new proofs of some old important
results concerning homogeneous Vaisman manifolds.

Theorem 4.1: ([Va2]) The canonical foliation of a compact homogeneous Vaisman
manifold $(M=G/H,I,g)$ is regular.111A foliation on a manifold is regular if
its leaf space is a manifold; and it is quasi-regular if all its leaves are
compact, in which case the leaf space is an orbifold.

Proof: Since $M$ is compact, $\Sigma$ has at least one compact leaf (this was
proven in [T], but follows also by results of Kato (in [K]) or [OV4]). Then,
by homogeneity, all leaves are compact, and hence $\Sigma$ is quasi-regular.
Consider the elliptic fibration $\pi:M\rightarrow M/\Sigma$. This map has at
least one smooth fibre. To see this, just consider $\pi$ with values in the
smooth part of $M/\Sigma$ and apply Sard-Brown’s theorem. But then, again by
homogeneity, all fibers are smooth.

Remark 4.2: On homogeneous Vaisman manifolds the foliation $\Sigma$ is
regular, but the foliation $\langle\theta^{\sharp}\rangle\subset\Sigma$
generated by the Lee field is not necessarily regular. Indeed, consider the
Hopf surface $H=\frac{({\mathbb{C}}^{2}\backslash 0)}{\langle A\rangle}$,
where $A=\alpha\operatorname{Id}$ and $\alpha$ is a complex number such that
$\frac{\alpha}{|\alpha|}$ is not a root of unity. In this case the Lee field
is a radial vector field on ${\mathbb{C}}^{2}$, and its trajectory applied to
a 3-dimensional sphere gives a diffeomorphism
$v{\>\longrightarrow\>}\frac{\alpha}{|\alpha|}$ which has infinite order.

A consequence of 4 is the following.

Corollary 4.3:  A compact homogeneous Vaisman manifold $M$ has $b_{1}=1$.

Proof: Since the canonical foliation $\Sigma$ is regular, $M$ is a
$T^{2}$-fibration over a homogeneous projective manifold $P$. Since any
algebraic group is rational, homogeneous projective manifolds are also
rational. Using [D, Chapter 4], we obtain that $P$ is simply connected.

Write the exact sequence

$0{\>\longrightarrow\>}H^{1}(P){\>\longrightarrow\>}H^{1}(M){\>\longrightarrow\>}H^{1}(T^{2})\stackrel{{\scriptstyle\gamma}}{{{\>\longrightarrow\>}}}H^{2}(P)$

and observe that the rank of $\gamma$ is 1 (see [OV1]). Therefore,
$b_{1}(P)=0$ implies that $b_{1}(M)=1$.

Remark 4.4: One can construct homogeneous Vaisman manifolds as in [Va2],
starting from a compact homogeneous projective manifold $P=G/H$ such that the
action of $G$ on $P$ can be linearized (id est $P$ admits an equivariant ample
line bundle). Note that the structure of compact homogeneous Kähler manifolds
is clarified in [M]: up to biholomorphisms, they are products of flag
manifolds.

Acknowledgment: We are grateful to Florin Belgun for pointing out an error in
the initial proof of 3; thanks to him, the actual proof is much simpler.

## References

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Liviu Ornea
University of Bucharest, Faculty of Mathematics and Informatics,
14 Academiei str., 70109 Bucharest, Romania, and:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Str. 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University Higher School of
Economics, 6 Usacheva Str. Moscow, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>


Effect of Interfacial Defects on the Electronic Properties of MoS2 Based
Lateral T-H Heterophase Junctions

Mohammad Bahmani1,†, Mahdi Ghorbani-Asl2,‡, and Thomas Frauenheim⋆,1,3,4

1 Bremen Center for Computational Materials Science (BCCMS), Department of
Physics, Bremen University, 28359 Bremen, Germany

2 Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany

3 Beijing Computational Science Research Center (CSRC), 100193 Beijing, China

4 Shenzhen JL Computational Science and Applied Research Institute, 518110
Shenzhen, China

The coexistence of semiconducting (2H) and metallic (1T) phases of MoS2
monolayers have further pushed their strong potential for applications in the
next generation of electronic devices based on the two-dimensional lateral
heterojunctions. Structural defects have considerable effects on the
properties of these 2D devices. In particular, the interfaces of two phases
are often imperfect and may contain numerous vacancies created by phase
engineering techniques, e.g. under the electron beam. Here, the transport
behaviors of the heterojunctions in the existence of point defects are
explored by means of first-principles calculations and non-equilibrium Green’s
function approach. While vacancies in semiconducting MoS2 act as scattering
centers, their presence at the interface improves the flow of the charge
carriers. In the case of $V_{Mo}$, the current has been increased by two
orders of magnitude in comparison to the perfect device. The enhancement of
transmission was explained by changes in the electronic densities at the T-H
interface, which open new transport channels for electron conduction.

†<EMAIL_ADDRESS>‡<EMAIL_ADDRESS>⋆ frauenheim$@$uni-
bremen.de

##  1 Introduction

Among the developing family of two-dimensional (2D) materials, transition
metal dichalcogenides (TMDs) provide one of the most diverse electronic
properties including topological insulators, semiconductors, (semi)metals and
superconductors [1, 2, 3]. Noticeably, such a difference in the electronic
structure of TMDs correlates with their structural configurations, called
phases [4]. Monolayers of MoS2 in H-phase, with trigonal prismatic
coordination of metal atoms is a semiconducting material [5, 6], while T-phase
with octahedral coordination shows metallic character. The H-phase monolayer
is reported to be a promising material for field-effect transistors (FETs)
with small-scale channel lengths and negligible current leakage [5, 6].

Recent experiments have already shown controlled transitions from one phase to
another via external stimuli such as electron beam [7], ion intercalation [8],
or laser irradiation [9]. These phase-engineered 2D materials with minimum
variations in atomic structure and uniform stoichiometry not only demonstrate
rich physical behavior but also open up new avenues for the design of
electronic devices. The fabrication of lateral metallic/semiconducting
heterostructures has been suggested as a practical method to minimize the
contact resistance at the interface between 2D semiconductors and metal
electrodes. In particular, the formation of covalent bonds between the two
phases can introduce paths for carriers to travel across the interfaces, thus,
the Schottky barrier and contact resistance are reduced [10, 11, 12, 13]. It
has also been demonstrated that 1T-phase engineered electrodes in MoS2 based
electronic devices would generate ohmic contacts and, as a result, improve
electrical characteristics [14, 12].

Apart from intrinsic defects, the local phase transitions induced by electron
beam irradiation may give rise to the formation of point defects, in
particular at the interface of the two phases. [15, 16, 17, 18, 19, 20, 21,
22]. Defects can also be intentionally introduced during the post-growth stage
via ion bombardment, plasma treatment, vacuum annealing, or chemical
etching.[15, 16, 17, 18, 19, 20, 21, 22]. Indeed, the theoretical and
experimental results showed that the presence of sulfur vacancies can decrease
the energy difference between the H and T phases and eventually stabilize the
1T phase in MoS2 monolayer [23, 24]. The presence of point defects in
semiconducting MoS2 monolayers leads to the observation of the localized
states in their electronic structure, which act as short-ranged scattering
centers for charge carriers [25, 26, 27, 28]. Hence, defects were found to
deteriorate the mobility of the fabricated devices [29, 30, 31]. It was also
shown that sulfur line vacancies in MoS2 can behave like pseudo-ballistic wire
for electron transport [32].

So far, several theoretical studies have reported the transport properties of
phase-engineered devices based on TMDs monolayers including MoS2 based lateral
junctions [33, 34, 35, 36, 11, 12]. In most of these studies, however, it is
assumed that two phases have a perfect crystalline structure and connected via
an atomically sharp and defect-free interface.

Here, transport properties of devices based on MLs MoS2, containing various
point vacancies and antisites at the interface between metallic and
semiconducting phases, are the subject of the present study. Our systematic
investigations show significant improvements in the current, as molybdenum
vacancy and vacancy complexes are created at the interfaces of two phases.
These findings render defect engineering as an efficient route to further
improve the performance of the devices based on the lateral heterojunctions
formed from TMDs.

##  2 Computational Details

Density-functional theory (DFT) calculations were performed using numerical
atomic orbitals (NAOs) basis sets as implemented in SIESTA code [37, 38]. The
norm-conserving pseudopotentials, including the effect of core electrons, are
employed, which were obtained using the Troullier-Martin method [39, 40]. The
Perdew-Burke-Ernzerhof (PBE) functional in the generalized gradient
approximation (GGA) is used to describe the exchange and correlation
interactions [41].

In the optimization calculations, the Brillouin zone (BZ) of supercells was
sampled using a 9$\times$1$\times$3 Monkhorst–Pack grid. In the electronic and
transport calculations, 5 and 29 k-points were used, respectively, along the
transverse direction. The conjugate-gradients (CG) method was applied to
optimize the lattice vectors and atomic positions of all the structures and
interfaces. The geometries were considered relaxed when the Hellman-Feynman
forces on each atom became smaller than 10 meV/Å. The energy cut-off of 450 Ry
is used in the framework of the real-space grid techniques to obtain Hartree,
exchange, and correlation energies. The Split-Norm was set to 0.16 and the
Energy-Shift of 0.02 Ry was chosen to determine the confinement radii. The
total energy convergency criteria ($\bigtriangleup E_{tot}$) is chosen to be
$10^{-5}$eV for k-points and $10^{-4}$eV for energy cut-off. When the
difference between two consecutive steps was less than $10^{-4}$eV, the total
energies in self-consistent field (SCF) cycles were considered converging.

The electron transport calculations were performed using non-equilibrium
Green’s functions (NEGF) techniques, as implemented in TranSIESTA and TBtrans
[42, 43]. The same basis sets as for the electronic calculations, namely SZP,
were employed for the transport calculations. The current through the
heterophase junction under a finite bias voltage was calculated within the
Landauer formula [44]

$I=\frac{2e}{h}\int
Trace\left[G_{C}^{\dagger}(E)\Gamma_{R}(E)G_{C}(E)\Gamma_{L}(E)\right]\left(f_{L}(E)-f_{R}(E)\right)dE.$
(1)

Where $G_{C}^{\dagger}(E)$ and $G_{C}(E)$ are the retarded and advanced
Green’s functions of the channel region. The effect of left (L) and right (R)
electrodes are projected onto the scattering region via their corresponding
self-energies, $\Gamma_{L}(E)$ and $\Gamma_{R}(E)$. The Fermi distribution of
$f_{L}(E)$ and $f_{R}(E)$ represent the available states for electrons in the
left and right electrodes. The transport calculations were performed at 300 K.

##  3 Results and Discussion

Depending on the edge orientation of monolayers, armchair and zigzag
interfaces can be realized. The armchair interfaces have been shown to be most
stable against buckling [34, 11]. They are also energetically more favorable
than connecting the zigzag terminated edges in the sulfur-rich limit [33, 34].
The recent theoretical study showed that the conductivity of the armchair
edges is higher than the zigzag interfaces due to the presence of metallic Mo
zigzag chains along the transport direction [11]. Accordingly, we consider the
armchair interface in the present study. In order to create Schottky contacts
at the interfaces, the semiconducting 2H-phase of MoS2 (channel region) is
sandwiched between two metal electrodes of 1T-MoS2, as shown in Fig. 1.

Figure 1: Upper: Schematic of the T-H heterophase junction of MoS2 monolayer.
Electrodes (only 1T-MoS2) and channel region (a combination of 1T- and
2H-MoS2) are highlighted with shaded red and black, respectively. The
interfaces are indicated with green dashed-lines. Lower: Optimized structures
of the point defects at the left interface of the devices are shown. The
complete defective devices are shown in Fig. S1 in the supplementary
information.

The size of the whole device is 117.50 Å along the transport direction (Z
axis) and 22.00 Å in the transverse direction (X axis), including the channel
with a length of 98.46 Å corresponding to 31 unit cells of MoS2. The channel
length is long enough to avoid artificial interactions between the two
electrodes. Also, it includes small adjacent portions of the 1T phase as
buffer layers to provide a computationally convenient configuration for
calculating self-energies at the boundaries.[45]. The periodic boundary
conditions were applied along the axis transverse to the transport direction.
A vacuum layer of 50 Å normal to the monolayers was considered, which prevents
interactions between adjacent supercells.

Several junctions composed of 1T and 2H of MLs MoS2 are considered without
defects (perfect) and when containing point defects in the phase boundaries,
as shown in Fig. 1. We considered 10 types of point defects, most of which
were observed in experiments [46, 16, 17, 19, 47, 20] and their stability were
analyzed by DFT calculations [15, 48, 27, 28]. We look at a sulfur and a
molybdenum vacancy, $V_{S}$ and $V_{Mo}$, a double sulfur vacancy from upper
and bottom layers $V_{2S-top}$, and the case of removing two atoms from the
upper sulfur layer and parallel to the interface, $V_{2S-par}$. Besides,
vacancy complexes of molybdenum and three sulfurs ($V_{Mo+3S}$) and six
sulfurs ($V_{Mo+6S}$) are also studied. Four antisites are also considered:
${Mo}_{S}$, ${Mo}_{2S-top}$, ${S}_{Mo}$, and ${2S-top}_{Mo}$. Our electronic
structure calculations indicate that 2H-MoS2 is a semiconductor with a bandgap
of 1.73 eV, while 1T-MoS2 has a metallic character (see Fig. S2). These
results are in agreement with previous theoretical reports at the same level
of theory [49, 48, 50, 6, 51]. Such studies have shown no difference or a
difference of $\%0.63$ between the lattice constant of the T- and H- phase of
MoS2 monolayers. Therefore, the same lattice parameters, namely 3.176Å, are
used for both phases. Such a phase transition can be seen as the collective
displacement of sulfur atoms while the stoichiometry of the materials is
preserved. The constructed lateral heterostructures with armchair edges are
optimized, as shown at the top of Fig. 1. It should be noted that the
optimization could not transform 1T into the 2H phase but induce some
distortions, indicating the activation barrier for the phase transformation is
higher than the relaxation of the boundary. In addition, the atomic network
can be subjected to strain as a result of defects in the phase boundary. Fig.
S3 shows the strain map, which is specified as the total displacements in all
three axes as compared to the perfect interface. It can be seen that the
largest change in the atomic positions occurs in the case of $V_{Mo+3S}$ at
the interfaces while the sulfur vacancies induce the smallest displacements
into the phase boundary.

###  3.1 Sulfur vacancies

In this section, we present the electronic and transport properties of T-H
heterophase junction containing interfacial sulfur vacancies; $V_{S}$,
$V_{2S-top}$, and $V_{2S-par}$. In Fig. 2, local density of states (LDOS) on
the atoms at the left interface of such devices are plotted at Bias $=0.00\,V$
and Bias $=1.40\,V$. In the following, the term ”interface” is used for a part
of the device, which consists of atoms from one layer of 1T-MoS2 and one layer
of 2H-MoS2. Due to the electronic states from T phase, the band gap in LDOS is
narrower than that for pristine 2H phase of MoS2. Comparing the two figures,
there is a shift in the energy, corresponding to half of the applied voltage.

Figure 2: (Color online) LDOS at the left interface of T-H heterophase
junction containing various sulfur vacancies; $V_{S}$, $V_{2S-top}$, and
$V_{2S-par}$. Energies are shifted with respect to their corresponding Fermi
energy.

The presence of defects introduces new states close to the Fermi level and
increases electron density at the interface, which is mainly contributed by
metal $d$ orbitals. It is evident that defect-associated states are more
pronounced in the case of V2s-par where the electron density is enhanced in
the vicinity of the Fermi level, including a peak at -0.2 eV. The results
showed that other types of sulfur vacancies have a negligible impact on the
electronic structure around the Fermi energy. It should be noted that the
sulfur vacancies in 2H-MoS2 monolayers act as scattering centers and
consequently diminish the transport properties [25, 26, 27, 28].

Figure 3:  (Color online) Transmission spectra for T-H heterophase junction of
MoS2 monolayer containing various sulfur vacancies at both interfaces at a)
Bias $=0.00\,V$ and b) Bias $=1.40\,V$. Energies are shifted with respect to
their corresponding Fermi energy. The insets show the change in the electronic
transmission channels at the top of the valance band. c) I-V characteristics
for the same devices. The inset shows the current around the threshold
voltage.

In order to elaborate the electron conduction dependency on the geometry of
contact between the T and the H phases, transmission spectra for the junction
without and with interfacial defects at two Bias, $0.00$ and $1.40\,V$, are
shown in Fig. 3a and 3b, respectively. Corresponding to the band gap of
2H-MoS2, there is no transmission at zero bias within an energy range of
$1.7\,eV$ around the Fermi level. A comparison between perfect interface and
those containing sulfur vacancies shows a growth in transmission probability,
suggesting the contribution of defect states in electrical transport.
Specifically, the transmission coefficients close to the valance band edge can
be increased to almost two times for the case of $V_{2S-par}$ vacancy. The IV
characteristics of the studied T-H heterophase junction are shown in Fig. 3c.
The junction displays a non-linear current-voltage similar to the
characteristics of a resonant tunneling diode. The energy mismatch between the
Fermi energy of the metallic 1T electrodes and the lowest unoccupied levels of
the 2H phase causes the presence of zero current and the need for threshold
voltage to produce finite current flow through the junction. The value of
threshold voltage was reduced from 1.0 V for the perfect interface to 0.75V
for the interface with divacancy. The appearance of defect-associated resonant
states in the transmission spectra within the voltage window changes the
current through the system, leading to an increase by an order of magnitude,
when $V_{2S-par}$ vacancy is present at the interfaces, as shown in Fig. 3c.

###  3.2 Molybdenum vacancies and vacancy complexes

We calculate the electronic and transport properties of the T-H heterophase
junction when molybdenum vacancy, $V_{Mo}$, and vacancy complexes $V_{Mo+3S}$
and $V_{Mo+6S}$ are present at the interface. The LDOS of the interface is
shown in Fig. 4 at Bias $=0.00\,V$ and Bias $=1.40\,V$. Here, the applied bias
has shifted the energies. The electronic structure of the interfaces with a
single Mo vacancy varies more than that of a single sulfur vacancy. In the
case of larger point defects, i.e. $V_{Mo+6S}$, the electronic structure shows
several resonant states around the Fermi level which are mainly formed by Mo
$4d$ states. The defect-induced changes in the electronic structure affect the
carrier injection through the junction. The transmission function (Fig. 5b&c)
at the top of the valance band shows a significant enhancement when vacancies
are introduced into the interfaces. Accordingly, the current is increased by
up to three orders of magnitude in comparison to the perfect interface. This
is due to an enhancement of carrier occupations near the Fermi level, which
leads to an increase in the transmission spectrum. As a result, the interfaces
with $V_{Mo+6S}$ vacancy demonstrate a threshold voltage of ≈0.5 V, half of
that for a perfect interface.

Figure 4: (Color online) LDOS at the left interface of T-H heterophase
junction containing molybdenum vacancy, $V_{Mo}$, and vacancy complexes as
$V_{Mo+3S}$ and $V_{Mo+6S}$. Energies are shifted with respect to their
corresponding Fermi energy. Figure 5: (Color online) Transmission spectra for
T-H heterophase junction of MoS2 monolayer containing molybdenum vacancy,
$V_{Mo}$, and vacancy complexes as $V_{Mo+3S}$ and $V_{Mo+6S}$ at both
interfaces at a) Bias $=0.00\,V$ and b) Bias $=1.40\,V$. Energies are shifted
with respect to their corresponding Fermi energy. The insets show the change
in the electronic transmission channels at the top of the valance band. c) I-V
characteristics for the same devices. The inset shows the current around the
threshold voltage.

To further investigate the transport behaviour at the interface, we also plot
the vector current for perfect systems and devices with $V_{Mo}$ at both
interfaces (Fig. S4). Vector current displays the direction and the amount of
current, coming from the left or right electrode, projected on each atom and
at a specific energy channel. While the perfect interface shows dominant
current scattering at the T-H boundary for low bias voltages, e.g. V=-0.5 V,
the currents are delocalized in the channel region with $V_{Mo}$ at the
interface, suggesting that electrons have been well transmitted from
electrodes to the channel region.

We have also studied the influence of defect concentrations on the transport
properties through the interfaces. Here, , we fixed the length of the channel
but varied its width, including the interfaces with a single $V_{Mo}$. Fig.
LABEL:fgr:IV_concen shows the difference in the conductance through devices
($\Delta G\,=\,G_{defect}-G_{perfect}$) as a function of the devices’ areas.
It is evident that the conductance reduces with decreasing the defect
concentration approaching the value of the perfect interface for zero-defect
density. Variation of channel widths has two different effects on the
transport properties: On one hand, the number of transport channels increases
with the width of the channel. On the other hand, for a constant number of
defect sites, increasing the channel width leads to a decrease in the carrier
densities around the Fermi Level. The former effect is canceled out by
subtracting the conductivity of the system from the corresponding one with the
pristine interface. As a result, for wider channels (lower concentrations),
the increase in electrical conductivity is linearly reduced.

Figure 6: (Color online) Difference in the conductance through perfect systems
and device with $V_{Mo}$ ($\Delta G\,=\,G_{defect}-G_{perfect}$) as function
of the device’s area at different bias voltages. Since number of vacancies is
the same for all cases, it is also a function of the defect density. It is
true that more transport channels are added to wider devices, however, at the
same time defect density is reduced, hence, decreasing the current through
defective devices. Thus, conductance difference is approaching to zero, as the
system gets wider.

###  3.3 Antisites

Because vacancy and antisite are both high electron-scattering centers, their
presence in TMDs can impair sample mobility [52]. We further investigate the
influence of antisites defects, such as $Mo_{S}$, $Mo_{2S-top}$, $S_{Mo}$, and
${2S-top}_{Mo}$, at the interfaces of T-H heterophase junctions. Among all the
considered antisites, the situation where molybdenum vacancy is substituted
with two sulfurs (${2S-top}_{Mo}$) provides the most pronounced defect
associated states at the valence band edge (Fig. 7). When compared with the Mo
vacancy, the defect states are more localized and originated mainly from
hybridization between the $Mo_{d}$-$S_{p}$ orbitals. The contributing orbitals
to the LDOS at the left interface are shown in Fig. S5 in the supplementary
information.

Figure 7: (Color online) LDOS at the left interface of T-H heterophase
junction containing various substitutions; $Mo_{S}$, $Mo_{2S-top}$, $S_{Mo}$,
${2S-top}_{Mo}$. Energies are shifted with respect to their corresponding
Fermi energy.

In Fig. 8a,b, transmission spectra for phase-engineered devices based on MLs
MoS2, containing various substitutions, are displayed at Bias
$=0.00\,\&\,1.40\,V$. Fig. 8c shows the corresponding IV characteristics as a
function of bias voltages up to $1.50V$. When vacancies are substituted with
sulfur or molybdenum atoms, the current stays in the same order as for the
device with perfect interfaces. Only for the case of ${2S-top}_{Mo}$, current
has been slightly increased due to the presence of midgap defect states
observed in LDOS and the enhancement in the transmission probabilities. It’s
also worth noting that, in contrast to the case of perfect interfaces, the
presence of antisite defects inevitably results in more phonon scattering
channels, which may be beneficial in lowering lattice thermal conductivity.

Figure 8: (Color online) Transmission spectra for T-H heterophase junction of
MoS2 monolayer containing various substitutions; $Mo_{S}$, $Mo_{2S-top}$,
$S_{Mo}$, ${2S-top}_{Mo}$, at both interfaces at a) Bias $=0.00\,V$ and b)
Bias $=1.40\,V$. Energies are shifted with respect to their corresponding
Fermi energy. The insets show the change in the electronic transmission
channels at the top of the valance band. c) I-V characteristics for the same
devices. The inset shows the current around the threshold voltage.

##  4 Conclusion

In the present paper, transport properties of charge carriers through devices
based on metallic (1T) and Semiconductor (2H) phases of MoS2 monolayers are
investigated. Various point defects are present at both interfaces: $V_{S}$,
$V_{2S-top}$, $V_{2S-par}$, $V_{Mo}$, $V_{Mo+3S}$, $V_{Mo+6S}$, $Mo_{S}$,
$Mo_{2S-top}$, $S_{Mo}$, and ${2S-top}_{Mo}$. The first-principles simulations
and NEGF technique are used to compute the LDOS, transmission curves, and IV
characteristics of perfect and defective devices under bias in the range of
$0.00\,V$ till $1.50\,V$. Our systematic study shows that defects at the
interfaces provide the opportunity for further improvement of the transport
properties of such devices. More notably, we found that transport properties
are enhanced in the presence of energetically favorable intrinsic point
defects. In contrast to the scattering character of defects in 2H-phase MoS2,
at the interface, they lead to emergence of resonant states close to the Fermi
level, thereby giving rise to the enhancement of the current flow. In
particular, creating a molybdenum vacancy induces defect midgap states in the
LDOS and improves the transport characteristics, which, in turn, leads to an
increase in the current up to two orders of magnitude. The knowledge developed
in this study could pave the way for the promising applications of lateral
heterojunctions of 1T/2H MoS2 monolayers in field effect devices.

## Author Contributions

Conceptualization, M.B. and M.G.-A; investigation, M.B.; methodology, M. B.,
and M.G.-A.; writing, M.B., M.G.-A., T. F.; supervision, M. G.-A, T. F. All
authors have read and agreed to the published version of the manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We thank the DFG funded research training group ”GRK 2247”.M.B. acknowledges
the support provided by DAAD and the PIP program at Bremen university. M.B.
also thanks Dr. Nick Papior for his support during the transport calculations
and Dr. Miguel Pruneda for his help to produce well-performed basis-sets and
pseudopotentials.

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