Investigation of Quark Distributions in a Family of Pentaquarks using the Thomas-Fermi Quark Model
Mohan Giri, Suman Baral, Gopi C. Kaphle, Nirmal Dangi, Sudip Shiwakoti, Leonardo Bardomero, Paul Lashomb, Walter Wilcox
IInvestigation of Quark Distributions in a Family ofPentaquarks using the Thomas-Fermi Quark Model
Mohan Giri a,b,c , Suman Baral b,d,e , Gopi Chandra Kaphle b,c , Nirmal Dangi b , SudipShiwakoti a,b , Leonardo Bardomero d , Paul Lashomb a , and Walter Wilcox a,b,da Department of Physics, Baylor University, Waco, TX USA 76798-7316 b Everest Institute of Science and Technology, 343 Ranibarimarg, Kathmandu, Nepal c Central Department of Physics,Tribhuvan University, Kirtipur 44618, Nepal d Neural Innovations LLC, 3225 Skinner Dr., Lorena, TX USA 76655 e Division of Health Science and Mathematics, Niagara County Community College,Sanborn, NY USA 14132August 7, 2020
Abstract
Using the Thomas-Fermi quark model, a collective, spherically symmetric density of states is createdto represent a gas of interacting fermions with various degeneracies at zero temperature. Over a familyof multi-pentaquarks, color interaction probabilities are obtained after averaging over all the possibleconfigurations. It is found that three different Thomas-Fermi functions are necessary for light, charm,and anti-charm quarks. These are assumed to be linearly related by proportionality constants resultingin consistency conditions. We analyze these conditions and find that they lead to an interesting patternof spherical quark distributions.
Over the last two decades the existence of multi-quark states such as pentaquarks and tetraquarkshave been confirmed through the efforts of the LHCb[1, 2, 3, 4], BESIII[5, 6], Belle[7, 8, 9, 10] and othercollaborations. One can expect even more to be discovered in the years to come. The measurements of thestate productions by the LHCb [4] were determined to be pentaquark states of quark flavor content uudc ¯ c .Specifically, the measured charmonium-pentaquarks were the P C (4312) + , P C (4440) + , and P C (4457) + .One of the standard theoretical methods to investigate such multi-quark states is Lattice QuantumChromodynamics (LQCD). As the quark content increases, however, LQCD becomes more computationallyexpensive and time-intensive. A great deal of effort on theoretical setup in terms of Wick contractions,operator selection, wave function smearing and analysis must be done. Each state must be investigatedseparately and no global picture emerges. Larger systems also require larger lattices. In order to investigatethe dynamics of such exotic states, the Thomas-Fermi (TF) statistical quark model was developed [11] asan inexpensive alternative. We have pointed out that it could be key to identifying families of bound states,rather than individual cases[12]. The TF quark model has previously been applied to systems of multi-quarksto investigate ground state properties of baryons[13, 14]. It has also been used to examine the bound states ofmulti-quark mesons[15]. The latter paper suggested the existence of several tetra, octa, and hexadeca quarkstates. Due to its timely experimental significance, in this paper we begin a theoretical investigation of thequark distributions for the family of multi-charmonium pentaquarks. These would include penta (5), deca(10), pentedeca (15),... collections of quarks and antiquarks, where we require one additional c ¯ c combinationper penta addition.The paper is organized as follows. We first develop a formalism to count the number of color interactionsin Section 2. In Section 3 we put flavor and color interactions together, convert a discrete system of particles1 a r X i v : . [ nu c l - t h ] A ug nto a continuous system, and obtain normalization conditions. After that, we introduce the TF statisticalquark model and obtain expressions for potential and kinetic energies in Sections 4. The energy is minimizedby varying the density of states and TF differential equations are obtained in Section 5. The consistencyconditions are then formulated in Section 6 for various quark contents and degeneracies. We examine theresulting radial distributions of three different types of quarks (light, charmed and anti-charmed) in Section7. Finally, we give conclusions and outline of further work in Section 8. The types of possible color interactions will be six in number, the same as in Ref. [15]; namely, color-colorrepulsion (CCR), color-color attraction (CCA), color-anti-color repulsion (CAR), color-anti-color attraction(CAA), anti-color-anti-color repulsion (AAR), and finally anti-color-anti-color attraction (AAA). The statis-tics, however, will differ from before. In this section, we will determine the average number of times a giveninteraction will occur, perform a cross check on the calculations, and finally see that the system is boundthrough residual color coupling alone even in the absence of volume pressure.
Due to color confinement, objects made from quarks must be SU (3) color singlet states in order to existas free particles. Color singlets can be achieved in five different ways, as shown in Table 1.Number Combinations for a color singlet Representation1 red + blue + green rgb r ¯ b ¯ g r ¯ r b ¯ b g ¯ g Table 1: Five different ways to obtain a color singlet.A pentaquark is a system of either four quarks and an anti-quark or four anti-quarks and a quark. Inorder for it to be a color singlet, other arrangements are not possible. This requirement for pentaquarksto be color singlets, and the various ways they can form color singlets affects the number of occurrences ofcolor interactions. For example, a system of ten particles could equally be two red, two blue, four greenand two anti-green; or two red, three blue, three green, one anti-blue and one anti-green. So, when we talkabout the occurrence of color interactions, there is some probability for each occurrence. We calculate suchprobabilities for a system of 5 η particles in this subsection. For convenience, we refer to a color singletconsisting of five quarks as a pocket . This means a system of 5 η particles have η pockets in total. Any pocketcould equally be one of the six possibilities as shown in the Table 2 below.Number Color singlet1 rbg + r ¯ r rbg + b ¯ b rbg + g ¯ g r ¯ b ¯ g + r ¯ r r ¯ b ¯ g + b ¯ b r ¯ b ¯ g + g ¯ g Table 2: Different ways a color singlet pocket could be formed.In this paper, we are investigating structures of a family of hidden charm multi-pentaquarks. So, oursystem of fermions can have only the first three pockets from the table. We will let x be the number of r ¯ rrbg pockets, y be the number of b ¯ brbg pockets, and z be the number of g ¯ grbg pockets so that,2 + y + z = η. (1)Since all particles in this system of 5 η particles can interact with each other through color, there will be atotal of 5 η (5 η − / η possible configurationsand, at any given time, the system would be found in one of those configurations. In order to give equalfooting to all the color combinations, we counted all the possible interactions across all configurations. Inthis distribution, there will be 3 η η (5 η − / Here, we count the number of occurrences for interactions between the same type of pockets. For example,when a r ¯ rrbg pocket interacts with another r ¯ rrbg pocket, as given in Table 3, we can have six CCR type ofinteractions, ten CCA type of interactions, and so on, as shown in Table 4.When x number of r ¯ rrbg pockets interact with each other, the arrangement in Table 4 gets repeated x ( x − / x ( x − / x ( x − / b ¯ brbg pocket inter-acting with another b ¯ brbg and a g ¯ grbg pocket interacting with another g ¯ grbg , both of which will yieldthe same number as counted using Table 3. So, the total number of times the CCR type interactionscan occur can be written as 6 ( x ( x −
1) + y ( y −
1) + z ( z − /
2, for the CCA type interactions we have10 ( x ( x −
1) + y ( y −
1) + z ( z − /
2, and so on. r ¯ r r b gr rr r ¯ r rr rb rg ¯ r ¯ rr ¯ r ¯ r ¯ rr ¯ rb ¯ rgr rr r ¯ r rr rb rgb br b ¯ r br bb bgg gr g ¯ r gr gb gg Table 3: Possible quark interactions between the same type of pocket.Interaction Number of TimesCCR 6CCA 10CAR 4CAA 4AAR 1AAA 0Table 4: Occurrence of interactions between the same types of pockets
Here, we count the number of occurrences of interactions between different types of pockets. For example,when an r ¯ rrbg pocket interacts with a b ¯ brbg pocket, as given in Table 5, we can have five CCR typeinteractions, eleven CCA type interactions, and so on. The complete list is shown in Table 6.When x number of r ¯ rrbg pockets interact with y number of b ¯ brbg pockets, y number of b ¯ brbg pockets inter-act with z number of g ¯ grbg pockets, and z number of g ¯ grbg pockets interact with x number of r ¯ rrbg pockets,CCR type of interaction occur 5 ( xy + yz + zx ) times, CCA type of interaction occur 11 ( xy + yz + zx ) timesand so on. 3 ¯ r r b gb br b ¯ r br bb bg ¯ b ¯ br ¯ b ¯ r ¯ br ¯ bb ¯ bgr rr r ¯ r rr rb rgb br b ¯ r br bb bgg gr g ¯ r gr gb gg Table 5: Possible quark interactions between two different types of pocket.Interaction Number of TimesCCR 5CCA 11CAR 6CAA 2AAR 0AAA 1Table 6: Occurrence of interactions for different types of pocket
The last case we need to consider are interactions within each pocket. We will first consider the pocket r ¯ rrbg . If we first consider r , it can interact with all of the other four quarks, namely, r ¯ r , rr , rb and rg . Ifwe then consider the interactions involving ¯ r , there are four possibilities, but only three unique interactionsthat we have not yet counted, namely, ¯ rr , ¯ rb and ¯ rg . Similarly, the two unique interactions involving thesecond r that have not been previously counted are rb and rg . Lastly, b interacting with g as bg is the onlyremaining unique interaction and we have now counted all of the interactions. The results are shown inTable 7 below. Interaction Number of TimesCCR 1CCA 5CAR 2CAA 2AAR 0AAA 0Table 7: Interaction within the same pocketExactly the same number of possibilities can be constructed for b ¯ brbg as well as for g ¯ grbg . Therefore,the total number of times CCR type of interaction can occur is η , CCA is 5 η and so on. Here, we have usedthe fact that x + y + z = η . In the next step we varied x from 0 to η and y from 0 to ( η − x ), thereby giving equal footing to all thecolor combinations and counted color interactions. E i gives the total number of occurrences of the i th colorinteraction out of 3 η η (5 η − /
2. Putting together what we obtained in previous sections, we have for E i :4 i = η X x =0 η − x X y =0 η ! x ! y !( η − x − y )! " ( x + y + z ) + (cid:18) x ( x − y ( y − z ( z − (cid:19) + ( xy + yz + zx ) . (2)Here, E i has been expressed in terms of vectors where the components of each of the column vectors denotethe calculations summarized in Tables 4, 6, and 7. The first term in E i corresponds to the interactions within a pocket , the second term corresponds to interactions between the same type of pockets , and the lastcorresponds to interactions between different pockets . As an example, the first component of E i tells us thetotal number of occurrences of the i th color interaction involving CCR which can are made up by CCRinteractions within a pocket, between the same type of pockets, and between different pockets.Employing Mathematica , Eq. (2) simplifies to E i = η − η (8 η − η − η (16 η − × η − η (4 η − × η − η (2 η + 1) η − η ( η − η − η ( η − . (3)Since E i is the total number of occurrences of the i th color interaction, the probability of the i th interactioncan be obtained by dividing Eq. (3) by the total number of possible interactions between the color combi-nations we’re interested in, 3 η η (5 η − /
2. In addition, we define a probability notation that will help usin the calculation of energies. In this new table, i and j refer to colors, P refers to probability of interactionwith no anti particle, P refers to interaction between one particle and one anti-particle, whereas P refers tointeraction between two anti-particles. We also divide by three to provide equal footing for each color. Inthis notation, we arrive at Table 8.Color probability symbol Probability value P ii η − η − P ij η − η − P ii η − η − P ij η + 1)45(5 η − P ii ( η − η − P ij η − η − i < j in P ij , P ij and P ij .5 .2 Cross check on our counting As a cross check on the probabilities, adding them up yields, X i (cid:54) j P ij + P ij + P ij = 1 , so the probabilities sum to one, as desired.We will also check our pentaquark model to ensure that it is indeed a color singlet. Let ~Q denote thetotal color charge of the quarks. By definition, we have, ~Q = η X i =1 ~q i . Squaring both the sides, ~Q · ~Q = η X i =1 q i + 2 X i = j ~q i · ~q j (4)= 5 η · g + 2 × η X i E i C i . (5)In the first term, q i · q j = g . In the second term, we have divided by 3 η to average over all the possibleconfigurations. Here, C i is the coupling constant of i th interaction. Using Mathematica , we find that, X i E i C i = − × η − g η. Therefore, using this value, ~Q · ~Q = 0 . Hence, we see that our model is, indeed, a color singlet.It should also be noted that, if we add the product of coupling and probability, we find that − g / (5 η − In the previous section, we examined the system of 5 η particles in terms of color. Now, we wish toexamine the same system in terms of flavor and then combine our results with the corresponding colorprobabilities we calculated earlier. After that, we convert the discrete system into a continuum density ofstates and, finally, obtain normalization conditions that will help us calculate system energies. Our multi-quark system consists of 5 η particles, where η is the number of pockets. In each pocketthere are four quarks and one antiquark. If N I and ¯ N I represent the number of flavors and the number ofanti-flavors with degeneracy factors g I and ¯ g I , respectively, then, X I g I N I = 4 η, (6) X I ¯ g I ¯ N I = η, (7)where I = 1 indicates light quarks and I = 2 indicates heavy quarks. Degeneracy factors g and ¯ g can takeon a value of one, two, three or four depending on whether it is a light or heavy quark, which will be furtherexplained in the results. 6 .2 Putting flavors and colors together Since there are 4 η quarks and η anti-quarks, we can expect 4 η (4 η − / η ( η − / η interactions between color and anti-color, all of whichadd up to 5 η (5 η − /
2. Interactions between two flavors can only be either CCA or CCR type, interactionsbetween two anti-flavors can only be either AAR or AAA, and interactions between a flavor and anti-flavorcan only be CAR or CAA. Using the equations above, we can summarize the interactions as those shown inTable 9. Interactions Number of TimesCCR & CCA X I g I N I × X J g J N J − ! AAR & AAA X I ¯ g I ¯ N I × X J ¯ g J ¯ N J − ! CAR & CAA X I,J ¯ N I N J ¯ g I g J Table 9: Interactions due to flavorBy assigning the flavors with color interaction probabilities, we can develop the expression for the potentialenergy. In the expression for potential energy in Eq. (26) of Ref. [15], there are terms related to CCR andCCA, terms for AAR and AAA, and one term for CAR and CAA type. They can simply be obtained bythe expansion of the terms displayed above and separated into same flavor and different flavor pieces.
The system we have so far described has been one of discrete particles and, in order to apply the TFmodel, we have to convert it into a continuous system. If n Ii ( r ) and ¯ n Ii ( r ) represent quark density of particlesand anti-particles with flavor index I and color index i , respectively, then Eq. (6) and Eq. (7) can be writtenas X i,I Z d r n Ii ( r ) = 4 η, (8)and X i,I Z d r ¯ n Ii ( r ) = η. (9)In the above equations, the degeneracy factors are already included in the quark densities n Ii ( r ) and ¯ n Ii ( r ).For a particular color index i , the above equations can be written as, X I Z d r n Ii ( r ) = 4 η , (10)and X I Z d r ¯ n Ii ( r ) = η . (11)Similarly, for a particular flavor index I , these equations become, X i Z d r n Ii ( r ) = N I g I , (12)and X i Z d r ¯ n Ii ( r ) = ¯ N I ¯ g I . (13)7 .4 Fermi-Dirac normalization For individual colors, Eq. (12) can be written as3 Z d r n Ii ( r ) = N I g I . (14)We will assume equal quark color content and drop the index i from this equation, remembering to sum overcolors later. This gives 3 Z d r n I ( r ) = N I g I . (15)Similarly, for anti-particles, we have, 3 Z d r ¯ n I ( r ) = ¯ N I ¯ g I . (16)We will now introduce Thomas-Fermi functions, f I ( r ) and ¯ f I ( r ), as f I ( r ) = ra × α s (cid:18) π n I ( r ) g I (cid:19) , (17)and ¯ f I ( r ) = ra × α s (cid:18) π ¯ n I ( r )¯ g I (cid:19) , (18)where a = (cid:126) / ( m c ) gives the scale, m is the mass of lightest quark, and α s = g / ( (cid:126) c ) is the strong couplingconstant. Note that g I and ¯ g I are degeneracy factors. Eq. (15) and Eq. (16) can now be written as (cid:18) α s a (cid:19) π Z r max dr √ r ( f I ( r )) = N I , (19)and (cid:18) α s a (cid:19) π Z ¯ r max dr √ r (cid:0) ¯ f I ( r ) (cid:1) = ¯ N I . (20)We will introduce a dimensionless parameter x such that r = Rx where, R = a × α s ! (cid:18) πη (cid:19) . (21)In terms of the dimensionless parameter x , Eq. (19) and Eq. (20) reduce to the following normalizationconditions: Z x max dx √ x ( f I ( x )) = N I η , (22)and Z ¯ x max dx √ x (cid:0) ¯ f I ( x ) (cid:1) = ¯ N I η . (23) In this section, we first introduce the TF model to show how the kinetic and potential energies areexpressed as a function of the density of states. We then use the interaction probabilities from previoussections to build the expression for the potential energy for a family of multi-pentaquarks. The Thomas-Fermi Statistical model is a semi-classical model introduced to many fermion systems. It treats particles as aFermi gas at T = 0. Despite it utilizing Fermi statistics, it is not fully quantum mechanical since it does not8ave a quantum mechanical wave function but, rather, a central function related to particle density. Thisfunction is determined by filling states up to the Fermi surface at each physical location. The key idea of theThomas-Fermi quark model is to express both the kinetic energy and the attractive and repulsive potentialenergy contributions as a simple function of quark density.The general expression for the kinetic energy is explained in Section 2 of Ref. [15] and is embodiedin Eq. (25) of that reference. In order to apply this expression to the quark system, we will introducenormalized degeneracy densities ˆ n Ii and ˆ¯ n Ii given byˆ n Ii = 3 n Ii N I , (24)and ˆ n Ii = 3 n Ii N I . (25)This new form of the quark density is helpful in correctly normalizing energies when continuum sources areused. When summed over flavors and colors, this yields the total kinetic energy ( T ) T = X i,I Z r max d r (cid:0) π (cid:126) N I ˆ n Ii ( r ) (cid:1) π (cid:126) m I ( g I ) + X i,I Z ¯ r max d r (cid:0) π (cid:126) ¯ N I ˆ¯ n Ii ( r ) (cid:1) π (cid:126) ¯ m I (¯ g I ) . (26)The procedure for determining the potential energy will mirror that of [15] in Eq. (26) with the exceptionof the new probabilities found for the multi-pentaquarks. We use the probabilities and flavor statistics ofthe previous section, giving U = 43 g X I N I ( N I − Z Z d rd r (cid:16)P i P ii ˆ n Ii ( r )ˆ n Ii ( r ) − P i Z Z d rd r | ~r − ~r | ¯ n I ( r ) n J ( r ) , (31)where we have switched to the single-color particle densities n I and ¯ n I with normalizations (15) and (16). In Section 4, we calculated the total energy of a family of pentaquarks. Now, we want to formallyminimize the energy by varying the particle densities, while keeping the quark number constant. This willgive us the differential equations we need to solve.Let’s introduce Lagrange undetermined multipliers λ I and ¯ λ I associated with the constraints,3 Z d r n I ( r ) = N I g I , (32)and 3 Z d r ¯ n I ( r ) = ¯ N I ¯ g I , (33)respectively. Then, the total energy becomes, E = X I Z r max d r (cid:0) π (cid:126) (cid:1) π (cid:126) m I ( g I ) (cid:0) n I ( r ) (cid:1) + X I Z r max d r (cid:0) π (cid:126) (cid:1) π (cid:126) ¯ m I (¯ g I ) (cid:0) ¯ n I ( r ) (cid:1) − g η − X I ( g I N I − g I N I Z Z d rd r | ~r − ~r | n I ( r ) n I ( r ) − g η − X I = J Z Z d rd r | ~r − ~r | n I ( r ) n J ( r ) − g η − X I,J Z Z d rd r | ~r − ~r | ¯ n I ( r ) n J ( r )+ X I λ I (cid:18) Z r max d r n I ( r ) − g I N I (cid:19) + X I ¯ λ I (cid:18) Z r max d r ¯ n I ( r ) − ¯ N I ¯ g I (cid:19) . (34)Once again, the purpose of adding these terms involving the Lagrange multipliers is to allow a minimizationof the total energy while keeping particle number fixed.The variation of the density δn I ( r ) in Eq. (34) gives, (cid:0) π (cid:126) (cid:1) π (cid:126) m I (cid:18) n I ( r ) g I (cid:19) = 18 g η − N I g I − N I g I Z r max d r | ~r − ~r | n I ( r )+ 18 g η − X I = J Z r max d r | ~r − ~r | n J ( r ) + 24 g η − X I Z r max d r | ~r − ~r | ¯ n J ( r ) − λ I . (35)10imilarly, variation of the density δ ¯ n I ( r ) in Eq. (34) gives, (cid:0) π (cid:126) (cid:1) π (cid:126) 14 ¯ m I (cid:18) ¯ n I ( r )¯ g I (cid:19) = 24 g η − X I Z r max d r | ~r − ~r | n J ( r ) − λ I . (36)We also know that, Z r max d r | ~r − ~r | n J ( r ) = 4 π (cid:20)Z r dr r n J ( r ) r + Z r max dr r n J ( r ) r (cid:21) . (37)Let us define ¯ α I ≡ ¯ m / ¯ m I as the ratio of mass of the lightest quark to the I th flavour quark. CombiningEqs. (17), (18), and (37) with Eq. (35), we obtain in terms of the dimensionless parameter x , α I f I ( x ) = − λ I g Rx + 6 η η − X I ¯ g I Z x dx √ x ¯ f I ( x ) + x Z x max x dx (cid:0) ¯ f I ( x ) (cid:1) √ x + 9 η η − ( ( N I g I − N I Z x dx √ x f I ( x ) + x Z x max x dx (cid:0) ¯ f I ( x ) (cid:1) √ x + X I = J g J Z x dx √ x f J ( x ) + x Z x max x dx (cid:0) ¯ f J ( x ) (cid:1) √ x ) . (38)Differentiating Eq. (38) twice, we get the first of two Thomas-Fermi differential equations, namely, α I d f I ( x ) dx = − η η − X I ¯ g I (cid:0) ¯ f I ( x ) (cid:1) √ x − η η − ( N I g I − N I ( f I ( x )) √ x + X I = J g J ( f J ( x )) √ x . (39)Similarly, combining Equations (17), (18), and (37) with (36) and using the dimensionless parameter x , wehave, ¯ α I ¯ f I ( x ) = − ¯ λ I g Rx + 6 η η − X J g J "Z x dx √ x ( f J ( x )) + x Z x max x dx ( f J ( x )) √ x . (40)Finally, differentiating Eq. (40) twice yields,¯ α I d ¯ f I ( x ) dx = − η η − X J g J ( f J ( x )) √ x . (41)Eq. (39) and Eq. (41) are Thomas-Fermi differential equations for our system of pentaquarks. Similar to the atomic model, the TF quark model assumes heavy particles in the central region and lightparticles spread outside of it. In the case of hidden charm multi-pentaquarks, the u and d quarks are lightparticles which can have larger radii while the c and ¯ c are the heavy quarks relative to the u and d quarks.In this paper, we will be investigating whether the c or the ¯ c will be found within the innermost radius.This can depend on several factors like color-coupling probabilities, strength of color-coupling, separationbetween particles, number of flavors, mass of flavors, and more.In this section, we first obtain the consistency conditions required for our model to give a single collec-tive density of states and then discuss how the parameters will be chosen to solve consistency conditionsnumerically. 11 .1 The consistency conditions For the heavy particles, the TF function is f ( x ) and for the heavy antiparticles ¯ f ( x ). For the lightquarks, the TF function is f ( x ) and for light anti-particles, ¯ f ( x ). Since we are not dealing with lightanti-particles, ¯ f ( x ) will be zero.The TF quark model creates a single, collective, spherically-symmetric density of states. So, we will as-sume that the TF fermi functions of all particles will be linearly related to each other by some proportionalityfactor k and ¯ k . In other words, f ( x ) = kf ( x ) , (42)¯ f ( x ) = ¯ kf ( x ) . (43)Using these TF functions in the TF differential equations, we obtain,¯ α d ¯ f dx = − η η − √ x h g ( f ( x )) + g ( f ( x )) i , (44) α d f dx = − η η − √ x ¯ g (cid:0) ¯ f (cid:1) − η η − √ x (cid:20) ( N g − N ( f ( x )) + g ( f ( x )) (cid:21) , (45)and α d f dx = − η η − √ x ¯ g (cid:0) ¯ f (cid:1) − η η − √ x (cid:20) ( N g − N ( f ( x )) + g ( f ( x )) (cid:21) . (46)Inserting Eqs. (42) and (43) into Eq. (44), (45), and (46), we obtain, d f ( x ) dx = Q ( f ( x )) √ x , (47)where, Q = − η η − α ¯ k (cid:16) g k + g (cid:17) , (48) Q = − η η − k (cid:18) ¯ g ¯ k + 34 ( N g − N k + 34 g (cid:19) , (49)and Q = − η η − α (cid:18) ¯ g ¯ k + 34 ( N g − N + 34 g k (cid:19) . (50)Eqs. (48), (49), and (50) are the consistency conditions. Here, k , ¯ k and Q are three unknowns which can becalculated using these three consistency conditions. The internal number parameters which enter the model are the particle state numbers N , N and theantiparticle state number ¯ N . In addition, there are the particle and antiparticle degeneracy factors g , g and ¯ g . These parameters satisfy the particle number constraints (6) and (7). Due to their masses beingnearly equal, we will assume the density functions of the u and d quarks to be the same. Furthermore, sincespin up and spin down states are distinguishable between flavors, this gives us four distinguishable particleswith the same mass. The degeneracy factor, g , for the light particle therefore will have a value from one tofour. Charm and anti-charm can have a maximum degeneracy of two because there is no other quark witha similar mass.In the following we will investigate the solution and interpretation of the consistency conditions in thespecial case of the ground state system for a given multi-pentaquark. We will therefore choose the maximumpossible values for degeneracy factors g , g and ¯ g for each value of η . We will also specialize to the equalheavy state and number situation: N = ¯ N and g = ¯ g . Let us analyze the situation.For η = 1, the quark combination is c ¯ cqqq , where q represents a light quark species. If one q is flavor u with spin up, another q is flavor u with spin down, and the last q is flavor d with either spin up or in spin12own state, all three light flavors are distinguishable. Therefore, there should be a degeneracy of three forthe light quarks. Similarly, c and ¯ c could either be spin up or spin down, so their degeneracy is one. Hence,we should have N = 1, g = 3, N = 1, g = 1, ¯ N = 1 and, ¯ g = 1. For η = 2, the quark combination is( c ¯ cqqq c ¯ cqqq ). In order to maximize the degeneracy factors, it is clear that the ground state is representedby N = 2, g = 3, N = 1, g = 2, ¯ N = 1 and, ¯ g = 2. For η = 3, the only state parameters which add upcorrectly are: N = 3, g = 3, N = 3, g = 1, ¯ N = 3, ¯ g = 1. For η = 4, the maximum degeneracy stateshould have g = ¯ g = 2. Then, maximizing the light quark degeneracy factor, we then expect the groundstate to be given by N = 3 and g = 4. For η = 5 the only possibility is N = 5, g = 3, N = 5, g = 1,¯ N = 5, and ¯ g = 1. Finally, for η = 6, we have the maximally degenerate state N = 6, g = 3, N = 3, g = 2, ¯ N = 3 and ¯ g = 2. The normalization condition from Eq. (22) and Eq. (23) can be written for heavy charm and anti-charmas, Z x max dx √ x ( f ( x )) = N η , (51)and Z ¯ x max dx √ x (cid:0) ¯ f ( x ) (cid:1) = ¯ N η . (52)In the family of multi-pentaquarks under consideration, charm and anti-charm are always equal in number,hence N = ¯ N . If we assume that ¯ x max < x max we may substitute ¯ f ( x ) = ¯ kf ( x ) for all x in Eq. (52).Thus, we find (¯ k ) / Z ¯ x max dx √ x ( f ( x )) = N η . (53)Since Eq. (51) and Eq. (53) have same right hand sides, it follows that the left hand sides should also beequal. Note that the function ( f ( x )) is non-negative as it represents the number density and further that x , ¯ x max and x max are positive numbers. This implies that the value of integration keeps decreasing as theupper limit of integration decreases. In other words, if ¯ x max < x max then consequently ¯ k > Mathematica . We used themass of charm and anti-charm from [15]; the masses of charm and anti-charm were 1553 MeV while the massof the light quark was 306 MeV. We then obtained real values for k and ¯ k , the results of which are tabulatedin Table 10 below. η N g N g ¯ N ¯ g k ¯ k k and ¯ k for various TF pentaquark states.We can see that ¯ k > η = 1. This implies that the anti-charm quark has the smallest radius for η = 1 while the charm has the smallest radius for η > 1. Furthermore, the value of ¯ k gets smaller with higherquark content. This suggests that the anti-charm spreads out further as the quark content of the systemincreases. However, a numerical statement will have to wait until an actual evaluation of the f ( x ) functioncan be made. Note also the slight nonmonotonic behavior in ¯ k for η = 4 , η = 4 being the only configuration listed with g = 4, and thus somewhat exceptional.It is also physically clear that the light quark TF function surrounds the charm and anti-charm regions.13igure 1: For η = 1, all three particles can be found inside the inner region represented by coordinate x .The middle region bounded by coordinate x and x is for heavy charm and light quarks but not anti-charm.The outer region between x and x is populated only by light quarks.Figure 2: For η = (2 , , , , 6) the region inside coordinate x is populated by all three particles. The middleregion bounded by coordinate x and x is for heavy anti-charm and light quarks but not charm. The outerregion between x and x is populated only by light quarks.14n our further investigations of this family of pentaquarks, we will need three TF functions for light quarks,heavy charms and heavy anti-charms. The position of x and ¯ x have to be chosen carefully, depending onthe number of quarks. Figs. 1 and 2 illustrate the idea. We have initiated research into hidden charm multi-pentaquark families using the TF statistical model.We were able to evaluate the various color interaction terms and obtain the probabilities of particle andantiparticle interactions. This allowed us to form the kinetic and potential energies subject to flavor normal-ization conditions. We then obtained the appropriate TF differential equations and solved the consistencyconditions for a number of multi-pentaquark ground states. We found that this required at least three TFfunctions with three different radii. We observed that for a pentaquark, the ordering of the quark radiiis as in Fig. 1 where the heavy charm antiquark is limited to the innermost region. As the quark contentincreases, however, the heavy charm, rather than the anti-charm, is limited to the center region, as in Fig. 2.Our evaluations were carried out to η = 6, i.e., for a state that is a combination of 6 pentaquarks.There are additional steps which need to be taken before the mass spectrum and radial structure ofmulti-pentaquark family states can be fully delineated. The complete kinetic and potential energies willneed to be developed and related to the TF functions f , f and ¯ f , with the understanding that ¯ x = x ingeneral. The volume energy needs to be added and the numerical job of minimizing the total energy whilesolving the TF equations still remains to be done. To bring the evaluations to the same level of completenessas in Ref. [13, 14], the spin energies also need to be calculated for the various degenerate states based upontheir nonrelativistic wavefunctions. Although this represents a considerable amount of additional work, weare encouraged by the consistent mathematical structure and physical picture that seems to be emerging,and we pause here before continuing on.We thank the Baylor University Research Committee, the Baylor Graduate School, and the Texas Ad-vanced SuperComputing Center for partial support. We would like to acknowledge Mr. Sujan Baral andMs. Pratigya Gyawali, CEO and COO respectively from Everest Institute of Science and Technology for initi-ating EVIST research collaborations. We thank Mr. Bikram Pandey, Mr. Shankar Parajuli and Mr. PraveshKoirala for partial calculations and other helpful considerations. We also acknowledge the Grant Office atNiagara County Community College as well as the National Science and Research Society for EducationalOutreach of Nepal. 15 eferences [1] Roel Aaij et al. Observation of J/ψp Resonances Consistent with Pentaquark States in Λ b → J/ψK − p Decays. Phys. Rev. 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