In this paper, we consider an exotic baryon (pentaquark) as a bound state of
two-particle systems composed of a baryon (nucleon) and a meson.We used a
baryon - meson picture to reduce a complicated five-body problem to a simple
two-body problems. The homogeneous Lippmann-Schwinger integral equation is
solved in configuration space by using one pion exchange potential. We
calculate the masses of pentaquarks $\theta_{c}(uudd\bar{c})$,
$\theta_{b}(uudd\bar{b})$.Comment: we need to checked our calculations again for mor confidenc

Babaghodrat, Shahnaz

Monemzadeh, Majid

Tazimi, Narges

English

arXiv

We consider an exotic baryon (pentaquark) as a bound state of two-body systems composed of a baryon (nucleon) and a meson. We used a baryon-meson picture to reduce a complicated five-body problem to simple two-body problems. The homogeneous Lippmann-Schwinger integral equation is solved in configuration space by using one-pion exchange potential. We calculate the masses of pentaquarksθc(uuddc¯)andθb(uuddb¯).</jats:p

M. Monemzadeh

N. Tazimi

Sh. Babaghodrat

Crossref

Research ArticleCalculating Masses of Pentaquarks Composed ofBaryons and MesonsM. Monemzadeh, N. Tazimi, and Sh. BabaghodratDepartment of Physics, University of Kashan, Ghotb Ravandi Boulevard, Kashan 87317-51167, IranCorrespondence should be addressed to N. Tazimi; nt physics@yahoo.comReceived 24 December 2015; Accepted 10 May 2016Academic Editor: Antonio EreditatoCopyright © 2016 M. Monemzadeh et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited. The publication of this article was funded by SCOAP3.We consider an exotic baryon (pentaquark) as a bound state of two-body systems composed of a baryon (nucleon) and a meson.We used a baryon-meson picture to reduce a complicated five-body problem to simple two-body problems. The homogeneousLippmann-Schwinger integral equation is solved in configuration space by using one-pion exchange potential. We calculate themasses of pentaquarks 𝜃𝑐(𝑢𝑢𝑑𝑑𝑐) and 𝜃𝑏(𝑢𝑢𝑑𝑑𝑏).1. IntroductionThere are two types of hadrons, baryons andmesons. Baryonsare equivalent to the bound states of three quarks, andmesonsare known to be the bound states of a quark and an antiquark.However, QCD describes mesons and baryons even with amore intricate structure. There are anomalous mesons suchas 𝑔𝑔𝑔, 𝑞𝑞𝑔, and 𝑞𝑞𝑞𝑞, as well as exotic baryons like 𝑞𝑞𝑞𝑔,𝑞𝑞𝑞𝑞𝑞, and so forth. Pentaquarks are baryons with at leastfour quarks and one antiquark. In exotic pentaquarks, theantiquark has a flavor different from the other four quarks.Exotic hadrons containing at least three valence quarksare being studied fairly extensively in modern physics.Although there are hundreds of ordinary hadrons, exotic oneshave not been found stable yet. However,QCDdoes not rejecttheir existence. Pentaquark 𝜃+, studied in photo productionexperiments [1, 2], is a prototype of exotic hadrons in lightand strong quark sector. Theoretically, hadronic reactionscontribute to 𝜃+ production more vividly than other types ofreactions.The quark model is commonly used to describe hadrons.In this model, mesons are described as 𝑞𝑞 and baryons asthree-quark composite particles. In a more microscopic view,QCD usually serves to describe the strong interaction.According to Lipkin [3, 4] and Gignoux et al. [5], amongpentaquarks, the five-quark anticharmed baryons of𝑃0= [𝑢𝑢𝑑𝑐𝑠] and 𝑝 = [𝑢𝑑𝑑𝑐𝑠] or similar antibeauty baryonsare the most bound.A lot of experimental evidence on the existence of exotichadrons has been found since 2003. Exotic hadrons’ quantumnumbers cannot be justified based on two- and three-quark bound states. Pentaquarks of 𝑞𝑞𝑞𝑞𝑞 form are examplesof exotic baryon states. Conjugation quantity of 𝐶 chargeis not an accurate quantum number for baryons, and allcombinations of total spin 𝐽 and parity 𝑃 can exist. However,an exotic baryon combination can be readily identified by itselectric charge 𝑄 and its strangeness 𝑆. Some evidence hasbeen reported during the last few years. For example, thepentaquark 𝜃++ was proven to exist in Hermes experimentin Hamburg, Germany [6, 7].For exotic baryons, we consider the following:𝜃+: the existence of this exotic baryon was predictedin chiral solution model [8]. It has 𝑆 = +1, 𝐽𝑃 = 1/2+,and 𝐼 = 0. It is a narrow light-mass particle of1540MeV.These attributes initially made 𝜃+ a subjectof experimental observation by LEPS [9]. The mostsuitable hadronic decay mode to identify it is 𝜃+ →𝐾0𝑝.𝜃𝑐and 𝜃𝑏: the existence of the bound exotic hadron𝜃𝑐was predicted through bound Skyrmion approach.This particle has a mass of 2650MeV and quantumHindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016, Article ID 6480926, 4 pageshttp://dx.doi.org/10.1155/2016/64809262 Advances in High Energy Physicsnumbers 𝐽𝑃 = 1/2+ and 𝐼 = 0. An experiment [10]showed a positive signal at a mass of about 3.1 GeV,but it was not confirmed later [11]. In strongly boundstates, the decay mode 𝐾+𝜋−𝜋−𝑝 is easy to identify.Likewise, the mass of 𝜃𝑏with the same quantumnumbers 𝐽𝑃 = 1/2+ and 𝐼 = 0 was predicted to be5207MeV.Thepossibleweak decaymode is𝐾+𝜋−𝜋−+𝜋++ 𝑝.𝜃𝑐𝑠: it is the five-quark state with 𝐽𝑃 = 1/2− and 𝐼 = 0.In a quark model which includes color-spin interac-tion, it can be bound and despite its strong decay,it becomes stable [12]. The mass dependent on themodel parameters is predicted to be 2420MeV. Thisstate was traced in the Fermilab E791 experiment via𝜙𝜋𝑝mode [13] and 𝐾∗0𝐾−𝑝mode [14].Lippmann-Schwinger equation for two-body boundstates is solved in Section 2. In Section 3, the procedure of thestudy is given and pentaquark masses are calculated.2. Lippmann-Schwinger Equation forTwo-Body Bound StatesIn this part, the binding energy of the entire system (pen-taquark) is calculated by numerical solution of homogeneousLippmann-Schwinger equation for each subsystem of boundmeson and baryon. Schrodinger equation for a two-bodybound state with the potential𝑉 runs as the following integralequation [16, 17]:𝜓𝑏(𝑟) = −𝑚4𝜋∫∞0𝑑𝑟󸀠𝑟󸀠2∫1−1𝑑𝑥󸀠⋅ ∫2𝜋0𝑑𝜙󸀠exp(−√𝑚 󵄨󵄨󵄨󵄨𝐸𝑏󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑟 − 𝑟󸀠󵄨󵄨󵄨󵄨󵄨)󵄨󵄨󵄨󵄨𝑟 − 𝑟󸀠󵄨󵄨󵄨󵄨𝑉 (𝑟󸀠) 𝜓𝑏(𝑟󸀠) ,(1)where 𝐸𝑏stands for the binding energy of the two-bodybound system (meson + baryon). When the interactionpotential is considered as 𝑟 function, (1) is of the followingeigenvalue form𝐾(𝐸𝑏)󵄨󵄨󵄨󵄨𝜓𝑏⟩ = 𝜆 (𝐸𝑏)󵄨󵄨󵄨󵄨𝜓𝑏⟩ . (2)𝜆 = 1 is the highest positive eigenvalue. The eigenvalueequation is solved through direct method [16–19] using aregularization method [20, 21]. To discretize the integrals,Gauss-Legendremethod [22] is employed. 200 grid points areconsidered for 𝑟 and 𝑟󸀠, whereas 100 mesh points are quiteenough for 𝑥󸀠. The kernel of integral equation is diagonizedto find 𝜆 = 1 in the eigenvalue spectrum. The energycorresponding to 𝜆 = 1 will be the system’s binding energy.3. Numerical Results and DiscussionWe consider a pentaquark as a bound state of a two-particlesystem formed by a baryon and a meson. The hadronicmolecular structure consisting of a baryon and a meson isshown in Figure 1.Pentaquark Hadronic moleculeqqqqqqqq qqFigure 1: Pentaquark.Interaction potential has an essential role in solvingthe eigenvalue equation (5). Different potentials have beenintroduced for meson-baryon interaction. Yukawa potential(screened Coulomb potential) is proposed as one of theappropriate ones [15, 23]. This study deals with exotic baryonstates created by a meson and a nucleon. The 𝜋 exchangepotential is among the most prominent meson exchangeforces.𝜋 is the lightest hadron that can be exchanged betweena meson and a nucleon. Therefore, we consider only 𝜋exchange, and 𝜌 and 𝜔 meson exchange will be elaboratedon in our subsequent works. One-pion exchange potential(OPEP) is of the following form [15]:𝑉𝜋(𝑟)={{{(?⃗?𝑁⋅ ?⃗?𝐻) (2𝑆12𝑉𝑇(𝑟) + 4?⃗?𝑁⋅ ?⃗?𝑙)𝑉𝑐(𝑟) , 𝑟 > 𝑟0,𝑉0, 𝑟 < 𝑟0,(3)where for 𝑟 > 𝑟0𝑉𝜋(𝑟) = (?⃗?𝑁⋅ ?⃗?𝐻) (2𝑆12𝑉𝑇(𝑟) + 4?⃗?𝑁⋅ ?⃗?𝑙)𝑉𝑐(𝑟)= (𝐼2− 𝐼2𝑁− 𝐼2𝐻) (2𝑆12𝑉𝑇(𝑟) + 𝐾2− 𝑆2𝑁− 𝑆2𝑙)𝑉𝑐(𝑟) ,(4)where𝑉𝑐(𝑟) =𝑔𝐻𝑔𝐴2𝜋𝑓2𝜋(𝑚2𝜋)𝑒−𝑚𝜋𝑟3𝑟, (5)𝑉𝑇(𝑟) =𝑔𝐻𝑔𝐴2𝜋𝑓2𝜋(𝑚2𝜋)𝑒−𝑚𝑟6𝑟(3𝑟2+3𝑟+ 1) , (6)𝑉0= −62.79 (MeV) or − 276 (MeV) , (7)where 𝑔𝐴, 𝑓𝜋, 𝑚𝜋, and 𝑔𝐻are axial coupling constant,pion decay constant, pion mass, and heavy-meson couplingconstant, respectively. 𝐼 is the total isospin of meson-nucleonsystem and𝑆12≡ 4 [3 (?⃗?𝑁⋅ ?̂?) (?⃗?𝑙⋅ ?̂?) − ?⃗?𝑁⋅ (?⃗?𝑙)] . (8)Inserting 𝐼𝑁(nucleon isospin), 𝐼𝐻(meson isospin), 𝑆𝑁(nucleon spin), 𝑆𝑙(the lightest quark’s spin in the meson),and 𝐾 = 𝑆𝑁+ 𝑆𝑙into the potential, we presented hadronicmolecular structure of two pentaquarks in Table 2.In this calculation we have ignored the tensor term𝑆12for studying the pentaquarks ground state [24]; hence,we used the central part of Yukawa potential. PentaquarkAdvances in High Energy Physics 3Table 1: Potential parameters used for the bound state of heavypentaquarks [15].Parameter Value𝑔𝐴1.27𝑓𝜋131MeV𝑔𝐻−0.59𝑚𝜋138MeV𝑚𝑁𝑖938.92MeV𝑚𝐵5279MeV𝑚𝐷1867MeVTable 2: Hadronic molecular structure of pentaquarks.Hadronic structure Pentaquark (meson + nucleon)𝜃𝑐(𝑢𝑢𝑑𝑑𝑐) 𝑢𝑢𝑑/𝑢𝑑𝑑, 𝑑𝑐/𝑢𝑐𝜃𝑏(𝑢𝑢𝑑𝑑𝑏) 𝑢𝑢𝑑/𝑢𝑑𝑑, 𝑑𝑏/𝑢𝑏binding energy is defined as the energy used when breakinga pentaquark into its components, that is, meson and baryon.We adopted the following constants for the bound state ofheavy pentaquarks from [15] (Table 1).Pentaquark masses are calculated according to𝑀(pentaquark) = 𝑚meson + 𝑚baryon + 𝐸𝑏. (9)In order to find the binding energy (𝐸𝑏), first we solveLippmann-Schwinger equation for the two-body system. Inthis approach, the kernel is diagnosed and the eigenvaluespectrum is identified (spin-spin interaction in the potentialand spin splitting are ignored). The data required includereduced mass of mesons and nucleons, proposed bindingenergy, potential coefficients, 𝑟0= 1 or 1.5 fm, and 𝑟-cutoff =20 fm. Table 3 shows the binding energies that we found.We adopted Yukawa potential from [15] and our calcu-lated energies are in good agreement with [15]. We presentpentaquark masses in Table 4.4. Results and DiscussionIn this paper, we solved Lippmann-Schwinger equation forpentaquark systems. We managed to obtain the bindingenergy and used it to calculate the masses of these systems.The pentaquark is considered as the bound state of a baryonand a heavy meson. We used a baryon-meson picture toreduce a complicated five-body problem to one simple two-body problem. InTable 4, we have listed our numerical resultsformasses of pentaquark systems, and pentaquarkmasses arecompared with the results obtained in [25–27].Our method is appropriate for investigating tetraquarksystems too. In our previous work, we investigated thetetraquark as the bound state of a heavy-light diquark andantidiquark. We used the diquark-antidiquark picture toreduce a complicated four-body problem to simple two-bodyproblems.We analyzed diquark-antidiquark in the framework ofa two-body (pseudopoint) problem. We made use of thepotential coefficients proposed by Ebert et al. We solvedTable 3: The calculated binding energies (MeV) for 𝜃𝑐(𝐷 meson)and 𝜃𝑏(𝐵 meson) for 𝐼 = 0. Column A: 𝑉0= −276MeV and 𝑟0=1 fm; column B: 𝑉0= −62.79MeV and 𝑟0= 1.5 fm.Pentaquark A B A [15] B [15]𝜃𝑐113.32 8.21 113.99 8.45𝜃𝑏140 15.1 139.48 15.46Table 4: Pentaquark calculated masses (MeV) for 𝐼 = 0.Pentaquark Mass (A) Mass (B) Mass in [25–27]𝜃𝑐2665.92 2790.82 2650𝜃𝑏6104.6 6209.71 6391Lippmann-Schwinger equation numerically for charmdiquark-antidiquark systems and found the eigenvaluesto calculate the binding energies and masses of heavytetraquarks with hidden charms [28].Competing InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper.AcknowledgmentsThis work is supported by University of Kashan under Grantno. 65500.6. The authors also express their greatest gratitudetoDr.M. R. Hadizadeh for his helpful technical comments onthe numerical solution of the LS integral equation and also forcomments that helped them to revise the text of the paper.References[1] T. Nakano, D. S. Ahn, J. K. 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Advances in High Energy Physics

Calculating Masses of Pentaquarks Composed of Baryons and Mesons

We consider an exotic baryon (pentaquark) as a bound state of two-body systems composed of a baryon (nucleon) and a meson. We used a baryon-meson picture to reduce a complicated five-body problem to simple two-body problems. The homogeneous Lippmann-Schwinger integral equation is solved in configuration space by using one-pion exchange potential. We calculate the masses of pentaquarks θc(uuddc¯) and θb(uuddb¯)

Calculating Masses of Pentaquarks Composed of Baryons and Mesons

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