# Calculation of Binding Energy and Wave Function for Exotic Hidden-Charm Pentaquark

Authors: F. Chezani Sharahi; M. Monemzadeh
Journal: Advances in High Energy Physics
 (2021)

---

## Abstract

In this study, pentaquarkPc4380 composed of a baryon Σc, and a D¯∗ meson is considered. Pentaquark is as a bound state of two-body systems composed of a baryon and a meson. The calculated potential will be expanded and replaced in the Schrödinger equation until tenth sentences of expansion. Solving the Schrödinger equation with the expanded potential of Pentaquark leads to an analytically complete approach. As a consequence, the binding energy EB of pentaquark Pc and wave function is obtained. The results EB will be presented in the form of tables so that we can review the existence of pentaquark Pc. Then, the wave function will be shown on diagrams. Finally, the calculated results are compared with the other obtained results, and the mass of observing pentaquark Pc and the radius of pentaquark are estimated.


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## Body

## 1. Introduction

In the first researches, the existence of multiquark states is illustrated in its simplest possible form, in which baryons are made of three fundamental quarks and mesons from a quark and an antiquark [1]. Indeed, from the point of view of the mathematics and physics, there was no QCD theorem opposing the existence of exotic multiquark. In gauge field theory, the QCD principle allows the existence of multiquarks and hybrids, which include quark and gluonic degrees of freedom [2].The search for the pentaquark and its probing has a long history. About ten years ago, in the study of pentaquark, a big progress was occurred when LEPS collaborations started a search with a claim to find out strong evidence of the pentaquark with the mass of about1.540GeV [3]. Then, many theoretical and experimental approaches were pursued, and many ideas were proposed in this field. For example, Zou and his colleagues suggested that the components of pentaquark can be included in nucleon. Since the heavy quarks play an important role in stabilization of multiquark systems and these play exactly the same role as the hydrogen molecule in QED [4–7], there are theoretical predictions about exotic hidden-charm pentaquark. In particular, the possibility of the existence of hidden-charm molecular baryons composed of an anticharmed meson and a charmed baryon was systematically studied within the one boson exchange model [3].In 2003, LEPS collaboration reported the evidence of pentaquarks state with content of quark uudds¯ having a very narrow width [8]. Unfortunately, this exotic flavor was not confirmed in subsequent experiments [9, 10]. In fact, the possible theoretical arguments are presented for the nonexistence of a stable pentaquark s in the references [11, 12]. Also, this mode has not been found with a light flavor yet. However, the baryons with light flavor may be able to have a significant pentaquark components [2].A decade ago, numerous researches were conducted around the world to find exotic particles. The result of these efforts was the observation of the mesonicX, Y, and Z particles in Belle, BESIII, BABAR, and LHCb. Some of these were considered as candidates for exotic states because they do not fit in a regular mesonic structure [1]. The common point of the exotic states is that all of these contain the heavy quarks and antiquarks.Due to the heavy quarks, the exotic states can be stable, and light modes can be combined with a regular mode [13]. This estimation is consistent with the fact that all of the exotic states have a hidden c or b which this is experimentally observed. If this claim is valid that the heavy components stabilize multiquark systems by particle physics scientists, it will be a natural proposal from their experimental colleagues to search for exotic states consisting clear quarks c and b, for example, bc¯qq′ [2].Here, one hidden-charm molecular baryon composed of an anticharmed meson and a charmed baryon has been studied. This can have one of the two flavors, i.e., symmetric6F or antisymmetric flavor 3¯F as shown in Figure 1. So, spin parity of s-wave charmed baryon is Jp=1/2+ or Jp=3/2+ for 6F and Jp=1/2+ for 3¯F. The pseudoscalar or vector anticharmed meson is made of s-wave anticharmed mesons. In Figure 1, hidden-charm molecular states composed of the anticharmed mesons and the charmed baryons are placed inside the green range [6].Figure 1
(color line) Thes-wave charmed baryons with Jp=1/2+ and pseudoscalar or vector s-wave charmed mesons which form the molecular baryons with double charm [6].Recently, LHCb collaboration has observed two resonance structuresPc4380 and Pc4450 with mass and width decay MPc4380=4380±8±29MeV, ΓPc4380=205±18±86MeV and MPc4450=4449.8±1.7±2.5MeV, ΓPc4450=39±5±19MeV in the invariant mass spectrum J/ψp from Λb→J/ψpK. According to the final state J/ψp, it is concluded that two observing states Pc cannot be isosinglet, and these consist of hidden-charm quantum numbers. Also, it is suggested that structures for each of the states Pc are considered for Pc4380, Σc2455D¯∗ and Pc4450, Σc∗2520D¯∗ [2, 3].Binding energy may be calculated analytically by solving Schrödinger equation with the expanded potential of Pentaquark. This approach offers advantages over numerical solution of the Schrödinger equation for pentaquark. First, all values for binding energy can be calculated, and this is done more accurately. Second wave function can be computed and presented graphically (ϕr versus r), from which useful particle characteristics and data could be extracted.In this work, this is described by analytical solution of pentaquarkΣcD¯∗ in four sections: where apart from the introduction in section 1, the potential of pentaquark Pc4380 is discussed in section 2. Then, using the potential of pentaquark, the Schrödinger equation is analytically solved in section 3. Finally, important conclusions are discussed in section 4.


## 2. The Potential of PentaquarkPc and Its Expansion

Pentaquark is considered to be composed of one baryon and one meson. For pentaquark, with such a structure, the potential is given as follows [2]:(1)VΣcD¯∗r=13gg1fπ∇2YΛ,mπ,rJ0G0,where the coupling constant g=0.59±0.07±0.01 is extracted from the width of D∗ [14, 15], g1=0.75,0.94,1.95 [6, 16]. Also, it is the mass of pion mπ=135MeV and pion decay constant fπ=132MeV [2]. The amount of phenomenological cutoff parameter is considered Λ=0.8GeV−2.5GeV [2, 12]. Finally, the YΛ,mπ,r is [2]:(2)YΛ,mπ,r=14πre−mπr−e−Λr−Λ2−mπ28πΛe−Λr.Now, we are calculating and expanding∇2Y:(3)∇2YΛ,mπ,r=mπ24πre−mπr−e−Λr−Λ3−mπ2Λ8πe−Λr=14πb0+b1r+b2r2+b3r3+b4r4+b5r5+b6r6+b7r7+b8r8+b9r9+⋯,where(4)b0=−m3+Λ32+m2Λ2;b1=m42−Λ42,b2=−m53!+Λ52×2!−m2Λ36×2!;b3=m64!−Λ62×3!+m2Λ44×3!,b4=−m75!+Λ72×4!−3m2Λ56×4!;b5=m86!−Λ82×5!+m2Λ63×5!,b6=−m97!+Λ92×6!−5m2Λ714×6!;b7=m108!−Λ102×7!+3m2Λ88×7!,b8=−m119!+Λ112×8!−7m2Λ918×8!;b9=m1210!−Λ122×9!+2m2Λ105×9!.


## 3. Solving the Schrödinger Equation for PentaquarkPc

To investigate the existence of one bound state of pentaquark, we solve the Schrödinger equation with calculated potential in previous section for pentaquarkPc.The radial Schrödinger equation for two-body systems is(5)d2dr2+2rddr−ll+1r2Rn,lr+2μℏE−VrRn,lr=0.Takingℏ=1 and changing the variable ϕr=rRn,lr, Eq. (5) becomes(6)d2dr2ϕr+2μE−Vr−ll+12μr2ϕr=0.By placing the potentialVr=VΣcD¯∗r and the expanded shape of ∇2Y in Eq. (6), an equation is expressed as follows:(7)d2dr2ϕr+2μE−C0−C1r−C2r2−C3r3−C4r4−C5r5−C6r6−C7r7−C8r8−C9r9+⋯−ll+12μr2ϕr=0,where(8)Cn=B1bn;n=0,1,⋯,9;B1=112πgg1fπ2J0G0.In Eq. (8), for ΣcD¯∗ with I=1/2,J=3/2, it is placing the numerical value of J0G0=1 product from literature [2].By considering the following, the proposed reply (cf. [17, 18]) for the differential Eq. (7) yields(9)ϕr=NreMr=rneMr.Differentiating second degree ofϕ in Eq.(9) gives(10)ϕ′′r=nn−1r−2+2nM′r−1+M′′+M′2rneM.Here, we are solving differential equation (7) by considering expansion Vr up to the 10th order for calculating binding energy EB of pentaquark. Hence, an approximation was attempted up to the 10th order, which not only expanded the potential behavior up to the 10th order similar to the potential behavior in equation (1) but also the resulting of binding energy that has sufficient precision compared to numerical literature [2, 19]. This is indicating the adequacy of approximation.Now, we can considered two position forϕr. The first ϕ0=0 is for n=1, and the second ϕ0=cte is considered for n=0 that cte means a constant. Thus, we study them in separate subsections.


### 3.1. The Positionϕ0=0

To consideredϕ0=0 then Mr yields [19]:(11)Mr=a1r2+a2r3+a3r4+a4r5+a5r6+a6r7+a7r8+a8r9+a9r10.Replacing Eq. (11) into Eq. (10) and comparing two Eqs. of (7) and (10), the following expression (Eq.(12)) for r−2 is obtained in terms of ll+1 in Eq.(8), as well as a system of 11 nonlinear equations expressed later:(12)nn−1=ll+1.In the base state, two values forn, namely, n=0 and n=1 are obtained that according to Eq. (9) and the condition ϕ0=0, the value n=0 could be unacceptable, and Nr=r will be. Therefore, ϕr and ϕ′′r are obtained as follows:(13)ϕr=reMr,ϕ′′r=2M′r−1+M′′+M′2reM.After replacing Eq. (11) in Eq. (13) and separately equal to the different powers of r, the following nonlinear equations are obtained:(14)6a1=−2μE−C0,12a2=2μC1,20a3+4a12=2μC2,30a4+12a1a2=2μC3,42a5+9a22+16a1a3=2μC4,56a6+20a1a4+24a2a3=2μC5,72a7+16a32+24a1a5+30a2a4=2μC6,90a8+28a1a6+36a2a5+40a3a4=2μC7,110a9+25a42+32a1a7+42a2a6+48a3a5=2μC8,132a10+36a1a8+48a2a7+56a3a6+60a4a5=2μC9.Here, by replacing the numerical values of constants for pentaquarkPc4380, Σc=2455MeV, and D¯∗=2008.32MeV, we obtained the binding energy of pentaquark [2]. In Tables 1 and 2, different values of binding energy from Eq. (14) for Pc4380 have been shown.Table 1
E
B
MeV
,
M
P
c
MeV for Pc4380 and g1=0.94.
Λ
MeV
g
=
0.51
g
=
0.59
g
=
0.67
800
(-63, 4400.32)
(-40, 4423.32)
(-17, 4446.32)
850
(-62, 4401.32)
(-34, 4429.32)
(-6.7, 4456.62)
900
(-59, 4404.32)
(-26, 4437.32)
—
1000
(-45, 4418.32)
(-0.11, 4463.21)
—
1100
(-22, 4441.32)
—
—Table 2
E
B
Me
V
,
M
P
c
MeV for Pc4380 and g1=0.75.
Λ
MeV
g
=
0.51
g
=
0.59
g
=
0.67
800
(-90, 4373.32)
(-73, 4390.32)
(-56, 4407.32)
850
(-95, 4368.32)
(-75, 4390.32)
(-53, 4407.32)
900
(-99, 4364.32)
(-74, 4389.32)
(-49, 4414.32)
1000
(-102, 4361.32)
(-67, 4369.32)
(-32, 4431.32)
1100
(-98, 4365.32)
(-51, 4412.32)
(-3.14, 4460.18)
1200
(-86, 4377.32)
—
—In the tables aboveMPc, is calculated as follows:(15)MPc=MΣc+MD¯∗+EB.According to the obtained values, it is observed that ing1=0.75, the results for pentaquark mass are much closer to the MPc4380. Also, to check the results, one of the main differences between this paper and the other references [2, 20] is that the acceptable results are obtained for the MPc4380 only in the 800MeV≤Λ≤1200MeV, and these cannot be found in Λ≥1200MeV.


### 3.2. The Positionϕ0=cte

Now, we consideredϕ0=cte; thus, Mr will be as follows [19]:(16)Mr=a1r+a2r2+a3r3+a4r4+a5r5+a6r6+a7r7+a8r8+a9r9.Here, replacing Eq. (15) into Eq. (10) and comparing it to Eq. (7), a similar equation is obtained by Eq. (12). In this position, i.e., ϕ0=cte, the value n=0 could be acceptable, and Nr=1 will be. Therefore, ϕr and ϕ′′r are obtained as follows:(17)ϕr=eMr,ϕ′′r=M′′+M′2eM.The same as before, by replacing Eq.(15) in Eq.(16), 10 nonlinear equations are obtained, and we have(18)2a2+a12=2μC0−E,6a3+4a1a2=2μC1,12a4+4a22+6a1a3=2μC2,20a5+8a1a4+12a2a3=2μC3,30a6+9a32+10a1a5+16a2a4=2μC4,42a7+12a1a6+20a2a5+24a3a4=2μC5,56a8+16a42+14a1a7+24a2a6+30a3a5=2μC6,72a9+16a1a8+28a2a7+36a3a6+40a4a5=2μC7,25a52+18a1a9+32a2a8+42a3a7+48a4a6=2μC8,36a2a9+48a3a8+56a4a7+60a5a6=2μC9.Also, in this position forPc4380, we obtained the binding energy of pentaquark and the numerical coefficients of wave function. Table 3 shows the amount of binding energy from Eq. (17) and Λ=800MeV for Pc4380.Table 3
E
B
MeV
,
M
P
c
MeV for Pc4380 and Λ=800MeV.
g
1
g
=
0.51
g
=
0.59
g
=
0.67
0.75
(-15.99, 4447.33)
(-15.80, 4447.52)
(-15.62, 4447.7)
0.94
—
(-15.47, 4447.85)
(-15.27, 4448.05)
1.95
—
(-14.01, 4449.32)
—To confirm the existence of pentaquark states, binding energy must be negative, i.e.,EB<0. Also, the total mass of particles contributing of pentaquark (i.e., the sum of baryon Σc and meson D¯∗ masses) in addition to binding energy is closer to the mass of pentaquark Pc4380. Here, the obtained results indicate that the binding energy ranging −102≤EB≤−0.11 for pentaquark Pc4380 is negative. Also, it conforms to calculated results in the literatures [2, 20–23]. As mentioned above, obtained results are acceptable to a great extent, and they could be considered a clear evidence for the existence of a bound of five-quark state.Figure2 demonstrates the wave function’s diagrams for pentaquark Pc in Λ=800MeV, at different values of g and g1. These charts tend to zero at the given value. As shown in the graphs, wave functions become zero in x~0.12−0.13, indicating that the maximum pentaquark radius ranges from 23.67 to 25.64 fm.Figure 2
P
c
4380 at r in Λ=800MeV for different values of g and g1.


## 3.1. The Positionϕ0=0

To consideredϕ0=0 then Mr yields [19]:(11)Mr=a1r2+a2r3+a3r4+a4r5+a5r6+a6r7+a7r8+a8r9+a9r10.Replacing Eq. (11) into Eq. (10) and comparing two Eqs. of (7) and (10), the following expression (Eq.(12)) for r−2 is obtained in terms of ll+1 in Eq.(8), as well as a system of 11 nonlinear equations expressed later:(12)nn−1=ll+1.In the base state, two values forn, namely, n=0 and n=1 are obtained that according to Eq. (9) and the condition ϕ0=0, the value n=0 could be unacceptable, and Nr=r will be. Therefore, ϕr and ϕ′′r are obtained as follows:(13)ϕr=reMr,ϕ′′r=2M′r−1+M′′+M′2reM.After replacing Eq. (11) in Eq. (13) and separately equal to the different powers of r, the following nonlinear equations are obtained:(14)6a1=−2μE−C0,12a2=2μC1,20a3+4a12=2μC2,30a4+12a1a2=2μC3,42a5+9a22+16a1a3=2μC4,56a6+20a1a4+24a2a3=2μC5,72a7+16a32+24a1a5+30a2a4=2μC6,90a8+28a1a6+36a2a5+40a3a4=2μC7,110a9+25a42+32a1a7+42a2a6+48a3a5=2μC8,132a10+36a1a8+48a2a7+56a3a6+60a4a5=2μC9.Here, by replacing the numerical values of constants for pentaquarkPc4380, Σc=2455MeV, and D¯∗=2008.32MeV, we obtained the binding energy of pentaquark [2]. In Tables 1 and 2, different values of binding energy from Eq. (14) for Pc4380 have been shown.Table 1
E
B
MeV
,
M
P
c
MeV for Pc4380 and g1=0.94.
Λ
MeV
g
=
0.51
g
=
0.59
g
=
0.67
800
(-63, 4400.32)
(-40, 4423.32)
(-17, 4446.32)
850
(-62, 4401.32)
(-34, 4429.32)
(-6.7, 4456.62)
900
(-59, 4404.32)
(-26, 4437.32)
—
1000
(-45, 4418.32)
(-0.11, 4463.21)
—
1100
(-22, 4441.32)
—
—Table 2
E
B
Me
V
,
M
P
c
MeV for Pc4380 and g1=0.75.
Λ
MeV
g
=
0.51
g
=
0.59
g
=
0.67
800
(-90, 4373.32)
(-73, 4390.32)
(-56, 4407.32)
850
(-95, 4368.32)
(-75, 4390.32)
(-53, 4407.32)
900
(-99, 4364.32)
(-74, 4389.32)
(-49, 4414.32)
1000
(-102, 4361.32)
(-67, 4369.32)
(-32, 4431.32)
1100
(-98, 4365.32)
(-51, 4412.32)
(-3.14, 4460.18)
1200
(-86, 4377.32)
—
—In the tables aboveMPc, is calculated as follows:(15)MPc=MΣc+MD¯∗+EB.According to the obtained values, it is observed that ing1=0.75, the results for pentaquark mass are much closer to the MPc4380. Also, to check the results, one of the main differences between this paper and the other references [2, 20] is that the acceptable results are obtained for the MPc4380 only in the 800MeV≤Λ≤1200MeV, and these cannot be found in Λ≥1200MeV.


## 3.2. The Positionϕ0=cte

Now, we consideredϕ0=cte; thus, Mr will be as follows [19]:(16)Mr=a1r+a2r2+a3r3+a4r4+a5r5+a6r6+a7r7+a8r8+a9r9.Here, replacing Eq. (15) into Eq. (10) and comparing it to Eq. (7), a similar equation is obtained by Eq. (12). In this position, i.e., ϕ0=cte, the value n=0 could be acceptable, and Nr=1 will be. Therefore, ϕr and ϕ′′r are obtained as follows:(17)ϕr=eMr,ϕ′′r=M′′+M′2eM.The same as before, by replacing Eq.(15) in Eq.(16), 10 nonlinear equations are obtained, and we have(18)2a2+a12=2μC0−E,6a3+4a1a2=2μC1,12a4+4a22+6a1a3=2μC2,20a5+8a1a4+12a2a3=2μC3,30a6+9a32+10a1a5+16a2a4=2μC4,42a7+12a1a6+20a2a5+24a3a4=2μC5,56a8+16a42+14a1a7+24a2a6+30a3a5=2μC6,72a9+16a1a8+28a2a7+36a3a6+40a4a5=2μC7,25a52+18a1a9+32a2a8+42a3a7+48a4a6=2μC8,36a2a9+48a3a8+56a4a7+60a5a6=2μC9.Also, in this position forPc4380, we obtained the binding energy of pentaquark and the numerical coefficients of wave function. Table 3 shows the amount of binding energy from Eq. (17) and Λ=800MeV for Pc4380.Table 3
E
B
MeV
,
M
P
c
MeV for Pc4380 and Λ=800MeV.
g
1
g
=
0.51
g
=
0.59
g
=
0.67
0.75
(-15.99, 4447.33)
(-15.80, 4447.52)
(-15.62, 4447.7)
0.94
—
(-15.47, 4447.85)
(-15.27, 4448.05)
1.95
—
(-14.01, 4449.32)
—To confirm the existence of pentaquark states, binding energy must be negative, i.e.,EB<0. Also, the total mass of particles contributing of pentaquark (i.e., the sum of baryon Σc and meson D¯∗ masses) in addition to binding energy is closer to the mass of pentaquark Pc4380. Here, the obtained results indicate that the binding energy ranging −102≤EB≤−0.11 for pentaquark Pc4380 is negative. Also, it conforms to calculated results in the literatures [2, 20–23]. As mentioned above, obtained results are acceptable to a great extent, and they could be considered a clear evidence for the existence of a bound of five-quark state.Figure2 demonstrates the wave function’s diagrams for pentaquark Pc in Λ=800MeV, at different values of g and g1. These charts tend to zero at the given value. As shown in the graphs, wave functions become zero in x~0.12−0.13, indicating that the maximum pentaquark radius ranges from 23.67 to 25.64 fm.Figure 2
P
c
4380 at r in Λ=800MeV for different values of g and g1.


## 4. Conclusions

In this article, the pentaquarkPc4380 system consisting of baryon Σc and D¯∗ meson has been considered. The obtained potential for pentaquark in reference [2] was presented and expanded. Then, expanded potential was replaced in the Schrödinger equation, and that was solved as a bound state of two-body systems. By solving this to analytically approach and according to the values of constants and cutoff, 10 nonlinear differential equations and binding energy EB of pentaquark Pc and wave function coefficients were obtained. The resulting EB and wave function were presented using tables and diagrams in the previous section, which could confirm the existence of a bound state of pentaquark Pc4380. Then, it is specified that the wave function plots tend to be zero at a given value. Therefore, the maximum radius of pentaquark Pc was found out which ranged from x=23.67fm to x=25.64fm. We observed that the calculated values matches with the findings of others regarding the mass of Pc4380. Also, the advantage of this paper lies in its method, which other references are numerically calculated, and this paper delivers the obtained results analytically. Hence, the results are more comprehensive and complete compared to them.



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