Exploring the electromagnetic properties of the Ξ cD sandΩ

2025-04-27 0 0 392.65KB 17 页 10玖币

侵权投诉

Exploring the electromagnetic properties of the Ξ(′,∗)

c¯

D∗

sand Ω(∗)

c¯

D∗

smolecular states

Fu-Lai Wang1,2,3,∗Si-Qiang Luo1,2,3,†Hong-Yan Zhou1,2,‡Zhan-Wei Liu1,2,3,§and Xiang Liu1,2,3,4¶

1School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China

2Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China

3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,

and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China

4Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, China

This paper presents a systematic investigation of the electromagnetic properties of the hidden-charm molec-

ular pentaquarks within the constituent quark model. Speciﬁcally, it focuses on two types of pentaquarks: the

Ξ(′,∗)

c¯

D∗

s-type pentaquarks with double strangeness and the Ω(∗)

c¯

D∗

s-type pentaquarks with triple strangeness. The

study explores various electromagnetic properties, including the magnetic moments, the transition magnetic

moments, and the radiative decay behavior of these pentaquarks. To ensure realistic calculations, the S-Dwave

mixing eﬀect and the coupled channel eﬀect are taken into account. By examining the electromagnetic proper-

ties of the hidden-charm molecular pentaquarks with double and triple strangeness, this research contributes to

the deeper understanding of their spectroscopic behavior. These ﬁndings form a valuable addition to the ongoing

investigation into the broader spectrum of properties exhibited by the hidden-charm molecular pentaquarks.

I. INTRODUCTION

Since the observation of the charmoniumlike state X(3872)

in 2003, numerous new hadronic states have been experimen-

tally observed, leading to extensive discussions about their

properties. These eﬀorts have signiﬁcantly enriched our un-

derstanding of the hadron spectroscopy [1–22]. Moreover,

these studies have been valuable in deepening our understand-

ing of the nonperturbative behavior of the strong interactions.

Among the various assignments proposed for the observed

new hadronic states, the molecular state explanation has

gained popularity. Notably, in 2015, the LHCb Collabora-

tion observed two Pcstates [23] and subsequently reported

two substructures, namely Pc(4440) and Pc(4457) [24], corre-

sponding to the previously observed Pc(4450) [23]. Further-

more, they discovered a new state, Pc(4312), through a de-

tailed analysis of the Λb→J/ψpK process [24]. The LHCb

experiment has provided strong experimental evidence for the

existence of the hidden-charm molecular pentaquarks of the

Σc¯

D(∗)type [25–31]. In subsequent years, LHCb reported

the evidence for Pcs(4459) [32] and observed PΛ

ψs(4338) [33].

These exciting experimental advancements have not only en-

riched the hidden-charm pentaquark family [34–71,73?–

80], but have also inspired theorists to investigate the hidden-

charm molecular pentaquarks of the Pcss(s)type [81–84]. In

recent years, the Lanzhou group has conducted extensive stud-

ies on the mass spectra of the hidden-charm molecular pen-

taquarks with double and triple strangeness. Speciﬁcally, their

investigations have focused on the Ξ(′,∗)

c¯

D(∗)

s[81] and Ω(∗)

c¯

D(∗)

[82] interactions. These studies have provided valuable in-

sights into the properties and characteristics of these exotic

hadronic states.

∗wangﬂ2016@lzu.edu.cn

†luosq15@lzu.edu.cn

‡zhouhy20@lzu.edu.cn

§liuzhanwei@lzu.edu.cn

¶xiangliu@lzu.edu.cn

Currently, the investigation of the properties of the hidden-

charm molecular pentaquarks remains a fascinating and sig-

niﬁcant research topic in hadron physics. It oﬀers valuable in-

sights for constructing a comprehensive family of the hidden-

charm molecular pentaquarks. The study of the electromag-

netic properties serves as an eﬀective approach to unveil the

inner structures of hadrons. A notable example is the success-

ful application of the constituent quark model in describing

the magnetic moments of the decuplet and octet baryons [85–

87], with corresponding experimental data available [88].

Given the importance of the electromagnetic properties,

it is crucial to investigate the electromagnetic characteristics

of the hidden-charm molecular pentaquarks. Some discus-

sions on the electromagnetic properties of the Σ(∗)

c¯

D(∗)-type

and Ξ(′,∗)

c¯

D(∗)-type hidden-charm molecular pentaquarks have

been conducted within the constituent quark model [89–92].

These studies shed light on the inner structures of the dis-

cussed hidden-charm molecular pentaquarks. However, it is

important to note that the exploration of the electromagnetic

properties for the hidden-charm molecular pentaquarks is still

in its early stages. Thus, further eﬀorts are required to obtain a

comprehensive understanding of the electromagnetic proper-

ties of various types of hidden-charm molecular pentaquarks.

In this study, our focus is on investigating the electromag-

netic properties of the hidden-charm molecular pentaquarks

with double strangeness, speciﬁcally the Ξ(′,∗)

c¯

D∗

s-type pen-

taquarks, as well as those with triple strangeness, namely

the Ω(∗)

c¯

D∗

s-type pentaquarks. These particular pentaquark

states were initially predicted in Refs. [81,82]. Within

the framework of the constituent quark model, we exam-

ine their magnetic moments, transition magnetic moments,

and radiative decay behavior. Our realistic calculations in-

corporate the eﬀects of the S-Dwave mixing and coupled

channels. By undertaking this investigation, we aim to en-

hance our understanding of the electromagnetic properties

of the hidden-charm molecular pentaquarks with double and

triple strangeness, thereby contributing to the comprehensive

knowledge of these intriguing exotic hadrons [81,82].

The structure of this paper is as follows. In Sec. II,

arXiv:2210.02809v6 [hep-ph] 8 Aug 2023

we provide a detailed explanation of the methodology em-

ployed for calculating the electromagnetic properties of the

hadronic molecules. Additionally, we present the electromag-

netic properties of the Ξ(′,∗)

c¯

D∗

smolecular states. In Sec. III, we

shift our focus to the electromagnetic properties of the Ω(∗)

c¯

D∗

molecular states. Finally, we oﬀer a concise summary of our

ﬁndings in Sec. IV.

II. THE ELECTROMAGNETIC PROPERTIES OF THE

Ξ(′,∗)

c¯

D∗

sMOLECULES

In this section, we thoroughly investigate the electromag-

netic properties of two molecular states: the Ξ′

c¯

D∗

sstate with

I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

sstate with I(JP)=1/2(5/2−)

[81]. Speciﬁcally, we analyze their magnetic moments, transi-

tion magnetic moments, and radiative decay behavior. These

investigations yield valuable insights into the inner structures

of these states, oﬀering signiﬁcant information in this regard.

A. The magnetic moments and the transition magnetic

moments of the Ξ(′,∗)

c¯

D∗

smolecules

In the context of the constituent quark model, the hadronic

magnetic moment encompasses two key components: the spin

magnetic moment and the orbital magnetic moment. Speciﬁ-

cally, when considering the z-component of the spin magnetic

moment operator for a given hadron, denoted as ˆµspin

z, it can

be mathematically represented as follows [85–87,89–125]:

ˆµspin

z=X

2Mj

ˆσz j,(1)

where ej,Mj, and ˆσz j denote the charge, the mass, and

the z-component of the Pauli spin operator of the jth con-

stituent of the hadron, respectively. When examining the

hadronic molecule comprised of a baryon and a meson, the

z-component of the orbital magnetic moment operator, de-

noted as ˆµorbital

z, can be expressed in the following manner

[89–94,96,98,100,104,105,108,116]:

ˆµorbital

z=µL

bm ˆ

= Mm

Mb+Mm

2Mb

+Mb

Mb+Mm

2Mm!ˆ

Lz,(2)

where the subscript bcorresponds to the baryon, while the

subscript mpertains to the meson. Furthermore, ˆ

Lzdenotes

the z-component of the orbital angular momentum operator

linking the baryon and the meson. In this study, the masses

of the S-wave charmed baryons and the S-wave charmed-

strange meson are extracted from the Particle Data Group [88]

for reference.

As extensively discussed in various references such as [85–

87,89–125], the magnetic moments of the hadrons (µH)

and the transition magnetic moments between the hadrons

(µH→H′) are frequently estimated by evaluating the expecta-

tion values of the z-component of the total magnetic moment

operator (ˆµz), which can be represented as

µH=⟨JH,JH|ˆµz|JH,JH⟩,(3)

µH→H′=⟨JH′,Jz|ˆµz|JH,Jz⟩Jz=Min{JH,JH′}.(4)

Here, ˆµz=ˆµspin

z+ˆµorbital

z, and H(′)stands for either the fun-

damental hadron or the compound hadron. In the realistic

calculations, the previous theoretical studies commonly em-

ploy the maximum value of the third component of the total

angular momentum quantum number for the hadron to deter-

mine the hadronic magnetic moment. Similarly, they consider

the maximum third component of the total angular momen-

tum quantum number of the lowest state of the total angu-

lar momentum to discuss the transition magnetic moment be-

tween the hadrons [85–87,89–125]. In our current study, we

adopt the same model and convention as previous theoretical

works for calculating the hadronic magnetic moments and the

hadronic transition magnetic moments [85–87,89–125]. In

order to provide a comprehensive analysis, it is necessary to

discuss the wave functions of the hadronic states under con-

sideration. These wave functions encompass various aspects,

including the color part, the ﬂavor part, the spin part, and the

spatial part. Regarding the color wave function, it is straight-

forwardly represented by the constant value 1, as the color

aspect is typically treated uniformly in our context. On the

other hand, the ﬂavor-spin wave function can be constructed

by taking into account the symmetry constraints imposed by

the system. Finally, the spatial wave function can be derived

by quantitatively studying the mass spectrum of the corre-

sponding hadron [92].

In the subsequent analysis, we delve into the magnetic mo-

ments and the transition magnetic moments of two speciﬁc

molecular states: the Ξ′

c¯

D∗

smolecule with I(JP)=1/2(3/2−)

and the Ξ∗

c¯

D∗

smolecular state with I(JP)=1/2(5/2−). To ac-

complish this, we employ three distinct scenarios: the single-

channel analysis, the S-Dwave mixing analysis, and the cou-

pled channel analysis. These scenarios allow us to explore

the inﬂuence of the S-Dwave mixing eﬀect and the coupled

channel eﬀect on the magnetic moments and the transition

magnetic moments of the Ξ(′,∗)

c¯

D∗

smolecular states. By em-

ploying the aforementioned procedures, we can elucidate the

respective contributions of these eﬀects to the magnetic mo-

ments and the transition magnetic moments of the molecular

states under investigation.

1. The single channel analysis

Firstly, we investigate the magnetic moments and the tran-

sition magnetic moments of the Ξ′

c¯

D∗

smolecular state with

I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolecule with I(JP)=

1/2(5/2−), while considering only the S-wave component.

The ﬂavor wave functions of these states, denoted as |I,I3⟩,

can be expressed as [81]



2,1

2+=|Ξ(′,∗)+

cD∗−

s⟩,

2,−1

2+=|Ξ(′,∗)0

cD∗−

s⟩,

where Iand I3represent the isospins and the isospin third

components of the Ξ(′,∗)

c¯

D∗

ssystems, respectively. Further-

more, the spin wave functions |S,S3⟩for these states can be

constructed using the following coupling scheme [81]

Ξ′

c¯

D∗

s:|S,S3⟩=X

SΞ′

c,S¯

D∗

CS S 3

2SΞ′

c,1S¯

D∗

s

2,SΞ′

c+1,S¯

D∗

sE,

Ξ∗

c¯

D∗

s:|S,S3⟩=X

SΞ∗

c,S¯

D∗

CS S 3

2SΞ∗

c,1S¯

D∗

s

2,SΞ∗

c+1,S¯

D∗

sE.

Here, Sand S3represent the total spins and the total spin

third components for the Ξ(′,∗)

c¯

D∗

ssystems, respectively. The

Clebsch-Gordan coeﬃcient Ce f

ab,cd is utilized in the coupling

scheme. Additionally, SΞ′

c,SΞ∗

c, and S¯

D∗

scorrespond to the

spin third components of Ξ′

c,Ξ∗

c, and ¯

D∗

s, respectively.

With the aforementioned setup, we can now proceed to

calculate the magnetic moments of the Ξ′

c¯

D∗

smolecule with

I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolecular state with I(JP)=

1/2(5/2−), such as

µI3=1/2

Ξ′

c¯

D∗

s|3/2−⟩=χ|1

2,1

2⟩|1,1⟩

Ξ′+

cD∗−

s

ˆµz

χ|1

2,1

2⟩|1,1⟩

Ξ′+

cD∗−

s

=µΞ′+

c+µD∗−

s.(5)

Here, χs

frepresents the spin and ﬂavor wave functions of the

hadron, while the superscript sindicates the spin wave func-

tion and the subscript fdenotes the ﬂavor wave function. Fur-

thermore, in the context of the single channel analysis of the

hadronic magnetic moment, the overlap of the relevant spatial

wave function is 1. For brevity, this factor is omitted in the

above expression.

To determine the magnetic moments of the Ξ′(∗)

cbaryons

and the ¯

D∗

smeson, we employ the constituent quark model.

Initially, let us deﬁne the ﬂavor and spin wave functions of

these particles. The ﬂavor wave functions can be expressed as

follows:

Ξ′(∗)+

c:1

√2(usc +suc),Ξ′(∗)0

c:1

√2(dsc +sdc),D∗−

s: ¯cs,

while their corresponding spin wave functions |S,S3⟩can be

expressed as

Ξ′

c:











2,1

2+=1

√6(2↑↑↓ − ↓↑↑ − ↑↓↑)



2,−1

2+=1

√6(↓↑↓ +↑↓↓ −2↓↓↑)

Ξ∗













2,3

2+=↑↑↑



2,1

2+=1

√3(↓↑↑ +↑↓↑ +↑↑↓)



2,−1

2+=1

√3(↓↓↑ +↑↓↓ +↓↑↓)



2,−3

2+=↓↓↓

D∗

s:









|1,1⟩=↑↑

|1,0⟩=1

√2(↑↓ +↓↑)

|1,−1⟩=↓↓

Here, the notations ↑and ↓denote the third components of the

quark spins, with values of +1/2 and −1/2, respectively.

Based on the ﬂavor and spin wave functions of the Ξ′(∗)

baryons and the ¯

D∗

smeson, we can proceed to calculate their

magnetic moments. As an example, let us deduce the mag-

netic moment of the Ξ′+

cbaryon as follows:

µΞ′+

c=*χ

√6(2↑↑↓−↓↑↑−↑↓↑)

√2(usc+suc)

ˆµz

√6(2↑↑↓−↓↑↑−↑↓↑)

√2(usc+suc)+

3µu+2

3µs−1

3µc.(6)

In this study, we adopt the following deﬁnition for the mag-

netic magneton of the quark: µq=−µ¯q=eq/2Mq, where

eqrepresents the charge of the quark and Mqdenotes the

constituent mass of the quark. Utilizing this deﬁnition, we

can derive the expressions for the magnetic moments of the

Ξ′(∗)

cbaryons and the ¯

D∗

smeson. For the numerical analy-

sis, we utilize the constituent quark masses Mu=0.336 GeV,

Md=0.336 GeV, Ms=0.450 GeV, and Mc=1.680 GeV

to quantitatively investigate the electromagnetic properties of

these discussed hadrons. These constituent quark masses are

sourced from Ref. [86] and are widely employed in studies

related to the magnetic moments of the hadronic molecular

states [90,92,98].

In Table I, we present the expressions and numerical re-

sults for the magnetic moments of the Ξ′(∗)

cbaryons and the

D∗

smeson. Our obtained results align with those reported in

previous works [86,104,107,117,126–129]. In this study,

the magnetic moments and the transition magnetic moments

of hadrons are expressed in units of the nuclear magneton

µN=e/2MPwith MP=0.938 GeV [88]. As shown in Ta-

ble I, the Ξ′+

cand Ξ′0

cbaryons exhibit distinct magnetic mo-

ments, while the magnetic moment of the Ξ∗+

cdiﬀers from

that of the Ξ∗0

c. This discrepancy arises from the notable dif-

ference in the magnetic magnetons between the up quark and

the down quark, namely, µu=1.862 µNand µd=−0.931 µN.

Moreover, the Ξ′0

cand Ξ∗0

cexhibit approximately equal mag-

netic moments.

Based on our obtained magnetic moments of the Ξ′(∗)

baryons and the ¯

D∗

smeson, we can get the numerical re-

sults of the magnetic moments of the Ξ′

c¯

D∗

smolecule with

I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolecular state with I(JP)=

1/2(5/2−). In Table II, we present the expressions and numer-

ical results of the magnetic moments of the Ξ′

c¯

D∗

smolecular

state with I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolecular state

with I(JP)=1/2(5/2−) when performing the single channel

analysis.

As presented in Table II, the magnetic moments of the

Ξ′

c¯

D∗

s|3/2−⟩molecule with I3=1/2, the Ξ′

c¯

D∗

s|3/2−⟩molecule

with I3=−1/2, the Ξ∗

c¯

D∗

s|5/2−⟩molecule with I3=1/2,

and the Ξ∗

c¯

D∗

s|5/2−⟩molecule with I3=−1/2 are −0.414 µN,

−2.275 µN, 0.472 µN, and −2.321 µN, respectively. Notably,

the magnetic moment of the Ξ′

c¯

D∗

s|3/2−⟩molecular state can

be obtained as the sum of the magnetic moments of the Ξ′

baryon and the ¯

D∗

smeson. Furthermore, the magnetic moment

of the Ξ′+

csigniﬁcantly diﬀers from that of the Ξ′0

c, resulting

in a distinct magnetic moment for the Ξ′

c¯

D∗

s|3/2−⟩molecule

TABLE I. The magnetic moments and the transition magnetic mo-

ments of the Ξ′(∗)

cbaryons and the ¯

D∗

smeson. The magnetic moment

and the transition magnetic moment are given in units of µN, where

µNdenotes the nuclear magneton. The expressions for the magnetic

moments and the transition magnetic moments are enclosed in the

square brackets in the second column.

Quantities Our work Other works

µΞ′+

c0.654 h2

3µu+2

3µs−1

3µci0.65 [126], 0.67 [127]

µΞ′0

c−1.208 h2

3µd+2

3µs−1

3µci−1.20 [128], −1.20 [127]

µΞ∗+

c1.539 µu+µs+µc1.51 [129], 1.59 [104]

µΞ∗0

c−1.254 µd+µs+µc−1.20 [117], −1.18 [107]

µD∗−

s−1.067 µc+µs−1.00 [117], −1.08 [127]

µΞ∗+

c→Ξ′+

c0.199 h√2

3(µu+µs−2µc)i0.17 [86], 0.16 [130]

µΞ∗0

c→Ξ′0

c−1.117 h√2

3(µd+µs−2µc)i−1.07 [117], −1.03 [117]

TABLE II. The expressions and numerical results of the magnetic

moments of the Ξ′

c¯

D∗

smolecular state with I(JP)=1/2(3/2−) and

the Ξ∗

c¯

D∗

smolecular state with I(JP)=1/2(5/2−) when only the S-

wave component is considered.

Physical quantities Expressions Values

µI3=1/2

Ξ′

c¯

D∗

s|3/2−⟩µΞ′+

c+µD∗−

s−0.414 µN

µI3=−1/2

Ξ′

c¯

D∗

s|3/2−⟩µΞ′0

c+µD∗−

s−2.275 µN

µI3=1/2

Ξ∗

c¯

D∗

s|5/2−⟩µΞ∗+

c+µD∗−

s0.472 µN

µI3=−1/2

Ξ∗

c¯

D∗

s|5/2−⟩µΞ∗0

c+µD∗−

s−2.321 µN

with I3=1/2 compared to that with I3=−1/2. Similarly, the

Ξ∗

c¯

D∗

s|5/2−⟩molecular state exhibits diﬀerent magnetic mo-

ments for various I3quantum numbers. In addition, the mag-

netic moments of the Ξ′

c¯

D∗

s|3/2−⟩molecule with I3=−1/2

and the Ξ∗

c¯

D∗

s|5/2−⟩molecule with I3=−1/2 are nearly the

same, owing to the close magnetic moments of the Ξ′0

cand

Ξ∗0

In addition to investigating the magnetic moments, we also

examine the transition magnetic moments between the Ξ′

c¯

D∗

molecular state with I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolec-

ular state with I(JP)=1/2(5/2−). The transition magnetic

moment between these two states can be determined using the

following expression:

µI3=1/2

Ξ∗

c¯

D∗

s|5/2−⟩→Ξ′

c¯

D∗

s|3/2−⟩

=χ|1

2,1

2⟩|1,1⟩

Ξ′+

cD∗−

s

ˆµz

χ√2

5|3

2,3

2⟩|1,0⟩+√3

5|3

2,1

2⟩|1,1⟩

Ξ∗+

cD∗−

=r3

5µΞ∗+

c→Ξ′+

c.(7)

Hence, the transition magnetic moment for the Ξ∗

c¯

D∗

s|5/2−⟩ →

Ξ′

c¯

D∗

s|3/2−⟩γprocess can be connected to that of the Ξ∗

c→

Ξ′

cγprocess. It should be noted that the spatial wave func-

tions of the initial and ﬁnal states may inﬂuence the transition

magnetic moment, and this aspect will be addressed in the

subsequent subsection. Next, we proceed to estimate the tran-

sition magnetic moment for the Ξ∗+

c→Ξ′+

cγprocess, which

can be obtained from the expression

µΞ∗+

c→Ξ′+

c=*χ

√3(↓↓↑+↑↓↓+↓↑↓)

√2(usc+suc)

ˆµz

√6(2↑↑↓−↓↑↑−↑↓↑)

√2(usc+suc)+

=√2

3(µu+µs−2µc).(8)

Table Ipresents the expressions and numerical values of the

transition magnetic moments for the Ξ∗+

c→Ξ′+

cγand Ξ∗0

c→

Ξ′0

cγprocesses. The obtained results from our analysis are in

good agreement with the theoretical predictions reported in

Refs. [86,117,130].

Based on the calculated transition magnetic moments for

the Ξ∗+

c→Ξ′+

cγand Ξ∗0

c→Ξ′0

cγprocesses, we can determine

the values of the transition magnetic moments between the

Ξ′

c¯

D∗

smolecule with I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

smolec-

ular state with I(JP)=1/2(5/2−). Speciﬁcally, we ﬁnd that

µI3=1/2

Ξ∗

c¯

D∗

s|5/2−⟩→Ξ′

c¯

D∗

s|3/2−⟩=0.154 µN,

µI3=−1/2

Ξ∗

c¯

D∗

s|5/2−⟩→Ξ′

c¯

D∗

s|3/2−⟩=−0.866 µN.

It should be noted that the magnitude of the transition mag-

netic moment for the Ξ∗0

c→Ξ′0

cprocess is signiﬁcantly larger

than that for the Ξ∗+

c→Ξ′+

cprocess [117,130]. Consequently,

the absolute value of µΞ∗

c¯

D∗

s|5/2−⟩→Ξ′

c¯

D∗

s|3/2−⟩with I3=−1/2 is

considerably greater than that with I3=1/2.

2. The S -D wave mixing analysis

And then, we conduct further investigations on the mag-

netic moments and the transition magnetic moments of the

Ξ′

c¯

D∗

smolecular state with I(JP)=1/2(3/2−) and the Ξ∗

c¯

D∗

molecule with I(JP)=1/2(5/2−) by considering the addi-

tional contribution from the D-wave channels. Our calcula-

tions encompass the following S-wave and D-wave channels

for the Ξ′

c¯

D∗

smolecular state with I(JP)=1/2(3/2−) and the

Ξ∗

c¯

D∗

smolecular state with I(JP)=1/2(5/2−) [81]

Ξ′

c¯

D∗

s|3/2−⟩:|4S3/2⟩,|2D3/2⟩,|4D3/2⟩,

Ξ∗

c¯

D∗

s|5/2−⟩:|6S5/2⟩,|2D5/2⟩,|4D5/2⟩,|6D5/2⟩.

Here, we adopt the notation |2S+1LJ⟩to denote the spin S, or-

bital angular momentum L, and total angular momentum Jof

the molecular state under consideration.

By considering the inﬂuence of the S-Dwave mixing ef-

fect, we can derive the magnetic moment and the transition

magnetic moment of the molecular states through the follow-

ing deductions

i,j

µAi→Aj⟨ϕAj|ϕAi⟩,(9)

i,j

µBi→Aj⟨ϕAj|ϕBi⟩,(10)

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ExploringtheelectromagneticpropertiesoftheΞ(′,∗)c¯D∗sandΩ(∗)c¯D∗smolecularstatesFu-LaiWang1,2,3,∗Si-QiangLuo1,2,3,†Hong-YanZhou1,2,‡Zhan-WeiLiu1,2,3,§andXiangLiu1,2,3,4¶1SchoolofPhysicalScienceandTechnology,LanzhouUniversity,Lanzhou730000,China2ResearchCenterforHadronandCSRPhysics,LanzhouUniversitya...

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Exploring the electromagnetic properties of the Ξ cD sandΩ

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