Test of the nonrelativistic c\bar{c} potential
aa r X i v : . [ h e p - ph ] F e b Test of the nonrelativistic c ¯ c potential Ulugbek Yakhshiev
1, 2, ∗ Department of Physics, Inha University, Incheon 22212, Korea Theoretical Physics Department, National University of Uzbekistan, Tashkent 700174, Uzbekistan (Dated: March 1, 2021)We analyze the charmonium states by testing a phenomenological nonrelativistic potential andpropose a new set of parameters. This new set of parameters are fixed using only the lowest lyingS-wave states of charmonia where the spin-orbit and tensor interactions will not contribute. Afterfitting the parameters we analyze the whole fine structure of charmonium states taking into accountthe spin-orbit and tensor interactions too. Calculations showed that the nonrelativistic potentialmodel with the phenomenologically defined parameters is indeed well approximation for describingthe charmonium states.
PACS numbers: 12.38.Lg, 12.39.Pn, 14.40.PqKeywords: Heavy-quark potential, charmonium.
I. INTRODUCTION
An applicability of any nonrelativistic potential modelduring the studies of heavy hadrons spectra can be wellchecked by reproducing the heavy quarkonium states.The best candidate for this role is the charmonium con-sisting one heavy quark Q and one heavy antiquark¯ Q [1, 2]. Analyzing the charmonium spectra one canestablish some interesting features. For that purposeone keeps in mind some facts in formulating the bet-ter approach. First of all, one can note the large valueof the current quark mass in the characteristic energyscale m Q ≫ Λ QCD . The second, the smallness of heavy-quark velocity v Q ≪ c inside the charmonia which canbe crudely estimated from the radial excitation energydifferences corresponding to the given quantum numbers.These two facts indicate that the relativistic effects couldbe taken into account as the systematic corrections (forexample, see review [3]).From other side, one can also note the smallness ofnonperturbative effects in the spectra of charmonia. Theauthors of Ref. [4] estimated contribution to the spin-independent heavy quark potential due to the nonpertur-bative dynamics in the framework of instanton vacuummodel of quantum chromodynamics (QCD) [5, 6]. Theyalso found the relatively small value of difference betweenthe current and constituent quark masses in comparisonwith the constituent or current quark mass [4]. The re-cent calculations [7, 8], based on the estimations above,showed that the contributions due to the nonperturba-tive dynamics also can be considered as the small cor-rections. In particular, the authors of Ref. [8] showedthat the instanton effects are the first order perturba-tive corrections. Nevertheless, in the Ref. [7] it was dis-cussed that the instantons may shed some light on theorigin of parameters of the potentials used in the phe-nomenological approaches [9, 10]. The studies performed ∗ Electronic address: [email protected] in Refs. [4, 7] may lead to the conclusion that, althoughit is very nontrivial at low energies, at the high energiesthe nonperturbative dynamics seems to be tightly hiddenbehind the confinement mechanism which is not yet fullyunderstood.All said above more or less explains the success ofphenomenological potential model [11] where the compli-cated and unknown dynamics is expressed in terms of theeffective values of phenomenological parameters. There-fore, one has readily a nonrelativistic Schr¨odinger ap-proach for describing the energy spectra of heavy quarko-nium. Technically, the quarkonium is very similar to theHydrogen atom, i.e. one can solve the one body problemin the given external potential field instead of consider-ing the two body relativistic system. The difference fromthe Hydrogen atom problem is only due to the nature ofinteractions and its corresponding range. Consequently,due to the strong nature of interactions the excitation en-ergies of charmonia will be much lager than the electronexcitation energies in the Hydrogen atom. The size of acharmonium also will be much less in comparison withthe Hydrogen atom size due to the short range nature ofinteractions. One can use these obvious facts in apply-ing a numerical method to the charmonium problem andeasily find the appropriate variational parameters of themodel.Starting the discussions of the heavy quark potentialone can note that the basic spin-independent central in-teraction between the quark and antiquark can be wellseparated into two parts. The first scalar exchange partis fully phenomenological because of an unknown confine-ment mechanism. The most popular choice for this inter-action is expressed as a linearly increasing potential dueto the area law of Wilson loop [12] for the heavy-quarkpotential. The second, the vector exchange part is due tothe perturbative one gluon exchange mechanism at shortdistances and in the lowest order has the Coulomb inter-action like form with the corresponding running couplingconstant. The spin-dependent parts of interactions canbe reproduced from the central potential in the frame-work of nonrelativistic expansions [13]. The correspond-ing model is called a nonrelativistic constituent quarkmodel.In the present work, we discuss the interesting featuresof the nonrelativistic constituent quark model and pro-pose the new set of parameters for describing the charmo-nium states on a basis of updated experimental data [15].While we are doing that, as an input we concentrate onlyto the minimal part of spectrum instead of consideringthe whole spectrum. After fitting the parameters in amost compact way we concentrate to the whole spectrumand analyze the applicability of nonrelativistic potentialmodel approach.The paper is organized in the following way. In thenext section II, we briefly repeat the main features of themodel and describe very shortly a variational approachto the problem. In the section III, the results from cal-culations will be presented and discussed. In the lastsection IV, we summarize our results and make the cor-responding conclusions.
II. Q ¯ Q POTENTIAL AND VARIATIONALMETHOD
In the simple constituent quark model, the total Q ¯ Q potential has the following standard form V Q ¯ Q ( r ) = V C ( r ) + V SS ( r )( S Q · S ¯ Q )+ V LS ( r )( L · S )+ V T ( r ) (cid:2) S Q · n )( S ¯ Q · n ) − S Q · S ¯ Q (cid:3) , (1)where S Q ( S ¯ Q ) spin of the quark (antiquark), L is rel-ative orbital momentum, S = S Q + S ¯ Q is total spin ofthe quarkonium system. We work in the center of massframe and, therefore, a radius vector r is given in termsof the relative coordinates r = r Q − r ¯ Q and n = r /r defines the unit vector in direction of the radius vector.In Eq. (1) V C ( r ), V SS ( r ), V LS ( r ) and V T ( r ) are central,spin-spin, spin-orbit and tensor potentials depending onthe relative distance between the quark and antiquarks.The central part of the potential in a nonrelativisticreduction employs the following “Coulomb+linear” form V C ( r ) = κr − α s r , (2)where κ is parameter of string tension and α s ( µ ) = 1 β ln( µ / Λ ) (3)is the strong running coupling constant at the one-looplevel. Its value is determined from the characteristicenergy scale µ corresponding to the problem. Further, β = (33 − N f ) / (12 π ) is the beta function at the one-loop level and Λ QCD is the dimensional transmutationparameter. The nonrelativistic expansion of Q ¯ Q in-teractions allows to relate the spin-dependent parts of the potential to the central part [13]. So, the spin-dependent interactions corresponding to the “vector one-gluon-exchange+scalar confinement” are given as V (P) SS ( r ) = 32 πα s m Q δ σ ( r ) , (4) V (P) LS ( r ) = 12 m Q (cid:18) α s r − κr (cid:19) , (5) V (P) T ( r ) = 4 α s m Q r , (6)where m Q is heavy quark mass. In practical calculations,the pointlike spin-spin interaction in Eq. (4) is “smeared”by using an exponential function of the form δ σ ( r ) = (cid:18) σ √ π (cid:19) e − σ r , (7)where σ is smearing parameter. In such a way Q ¯ Q poten-tial is described in terms of only four parameters, κ , α s , m Q and σ . Usually, these parameters are found by fittingthe whole charmonium spectrum. In the present work wewill follow the phenomenological approach but find thoseparameters by fitting only some minimal part of S-wavespectrum instead of considering the whole spectrum.After fitting the form of potential, in order to eval-uate the energy states of quarkonia in a nonrelativisticpotential approach, one needs to solve the Schr¨odingerequation( ˆ H − E ) | Ψ JJ i = 0 . (8)Here ˆ H is Hamilton operator and | Ψ JJ i represents thestate vector with the total angular momentum J and itsthird component J . The coordinate space projection ofthe state vector h r | Ψ JJ i will reproduce the coordinatespace representation of the Hamiltonianˆ H ( r ) = − ~ m Q ∇ + V Q ¯ Q ( r ) , (9)where m Q arises from the doubled reduced mass of thequarkonium system. The matrix elements of Q ¯ Q poten-tial in the standard basis | S +1 L J i , which is given interms of the total spin S , the orbital angular momentum L , and the total angular momentum J satisfying the re-lation J = L + S , has the following form V Q ¯ Q ( r ) = h S +1 L J | V Q ¯ Q ( r ) | S +1 L J i = V ( r ) + (cid:20) S ( S + 1) − (cid:21) V SS ( r )+ h L · S i V LS ( r ) + (cid:26) − h L · S i ( h L · S i + 2)(2 L − L + 3)+ S ( S + 1) L ( L + 1)3(2 L − L + 3) (cid:27) V T ( r ) , (10)where h L · S i is defined as h L · S i = 12 [ J ( J + 1) − L ( L + 1) − S ( S + 1)] . The corresponding radial part of the wave functionfor a given orbital momentum L is a solution of theSchr¨odinger equation (cid:18) − ~ m Q ∇ + V Q ¯ Q ( r ) − E (cid:19) ψ LL ( r ) = 0 , (11)where an angular part of the wave function ψ LL ( r ) isrepresented in terms of the standard spherical harmon-ics Y LL (ˆ r ). In order to solve Eq. (11) numerically, wewill follow the gaussian expansion method (for details,see review [14]), where the state vector | ψ LL i is ex-panded in terms of a set of basis vectors {| φ nLL i ; n =1 , , . . . , n max } as | ψ LL i = n max X n =1 C ( L ) n | φ nLL i . (12)Here n is a radial quantum number. So, the radial ex-citations corresponding to the given angular momentumvalue will be reproduced naturally. In the gaussian ex-pansion method, the radial part φ GnL ( r ) of the total eigen-function in the spherical coordinate basis φ nLL ( r ) = φ GnL ( r ) Y LL (ˆ r ) (13)is expressed in terms of gaussian trial functions φ GnL ( r ) = (cid:18) L +7 / r − L − n √ π (2 L + 1)!! (cid:19) / r L e − ( r/r n ) . (14)For the given set n runs the values n = 1 , , . . . , n max and the corresponding r n ’s are playing the role of varia-tional parameters. The variational parameters could beoptimized using a geometric progression [14] r n = r a n − , n = 1 , , . . . , n max (15)and, therefore, the actual number of parameters is re-duced to the three (e.g. r , r n max and n max ) for the givenvalues of the orbital quantum number L , the spin S andthe total angular momentum J .The expansion coefficients C ( L ) n in Eq. (12) andthe eigenenergies E ( L ) n are determined by employingRayleigh-Ritz variational principle. This leads to a gen-eralized matrix eigenvalue problem n max X n =1 (cid:16) K ( L ) mn + V ( L ) mn − E ( L ) n N ( L ) mn (cid:17) C ( L ) n = 0 , (16) m = 1 , , . . . , n max , where the corresponding matrix elements are defined inthe following way K ( L ) mn = h φ mLL (cid:12)(cid:12)(cid:12)(cid:12) ˆ p m Q (cid:12)(cid:12)(cid:12)(cid:12) φ nLL i , (17) V ( L ) mn = h φ mLL | V Q ¯ Q | φ nLL i , (18) N ( L ) mn = h φ mLL | φ nLL i . (19) III. RESULTS AND DISCUSSIONS
As we mentioned above in the phenomenological ap-proaches the parameters of the model, κ , α s , σ and m c ,are fitted to the spectra of experimentally known char-monium states. For example, the authors of Ref. [11]proposed the set of potential parameters given in Table I(see the model referred as NR). Using that set of parame- TABLE I: Parameters of the nonrelativistic potential models.NR corresponds to the potential model in Ref. [11] where eleven charmonium states are used as an input, NR4 describesthe present work with the potential parameters correspondingto the four charmonium states as an input, respectively.The m c α s κ σ model [GeV] [GeV] [GeV ] [GeV]NR 1.4794 0.5461 0.1425 1.0946NR4 1.4796 0.5426 0.1444 1.1510 ters they calculated all allowed E1 radiative partial widthand some important M1 width. As an input for the fit-ting of parameters they used 11 meson states: 6 statescorresponding to the S-wave, 3 states corresponding tothe P-wave and 2 states corresponding to the D-wave,respectively. The input values of these energy states aregiven in Table II (see the 2 nd column). The results oftheir calculations showed that the nonrelativistic poten-tial model with the certain set of parameters describesthe charmonium spectrum very well.However, nowadays the experimental data is improvedand some new states where fixed in the particle data [15].The values of charmonium states extracted from the cur-rent experimental data are given in the last column of theTable II. The natural question arises – “How the param-eters of the nonrelativistic potential model will changeif one concentrates to the updated experimental data?”Partially, our aim in the present work is to answer thisquestion. However, our main aim in the present work isnot only fitting the updated experimental data by meansof the new set of parameters. In addition to the fittingprocess we want to check “How well does a nonrelativisticexpansion work in the potential approaches?”As we said above the authors of Ref. [11] fitted the pa-rameters of the potential to the all eleven , experimentallyknown at that time, states of the charmonium spectrum.Therefore, a beauty of nonrelativistic expansion seems re-mained to be hidden behind. We want to emphasize that,in principle, one can concentrate to the part of spectrumin order to fit the parameters of model. For example,one can concentrate to the S-wave part of spectrum forfitting the parameters of model. During this process thespin-orbit and tensor interactions are not contributing tothe total interaction. One can also act in opposite formby concentrating to the part of spectra where the spin-orbit and tensor interactions are important. The reasonfor the possibility of such choices is due to the fact thatthe central and spin dependent parts of the potential are TABLE II: Experimental and calculated spectrum of c ¯ c states.All energy states are given in MeV and the output resultsare rounded up to 1 MeV. Authors of NR model in Ref. [11]used 11 input states and their values are shown in the secondcolumn. In the present work in order to reproduce NR4 resultsonly 4 states are used as an input, and their values are shownin the fourth column.State Ref. [11] This work Exp. [15]Input NR Input NR4 J/ψ (1 S ) 3097 3090 3097 3098 3096 . ± . η c (1 S ) 2979 2982 2984 2984 2983 . ± . ψ (2 S ) 3686 3672 3686 3682 3686 . ± . η c (2 S ) 3638 3630 3638 3638 3637 . ± . ψ (3 S ) 4040 4072 4084 4039 ± η c (3 S ) 4043 4055 ψ (4 S ) 4415 4406 4422 4421 ± η c (4 S ) 4384 4397 χ c (1 P ) 3556 3556 3559 3556 . ± . χ c (1 P ) 3511 3505 3505 3510 . ± . χ c (1 P ) 3415 3424 3415 3414 . ± . h c (1 P ) 3516 3524 3525 . ± . χ c (2 P ) 3972 3978 3927 . ± . χ c (2 P ) 3925 3937 χ c (2 P ) 3852 3864 3862 +26+40 − − h c (2 P ) 3934 3945 χ c (3 P ) 4317 4325 χ c (3 P ) 4271 4293 χ c (3 P ) 4202 4227 h c (3 P ) 4279 4293 ψ (1 D ) 3806 3816 ψ (1 D ) 3800 3807 3822 . ± . ψ (1 D ) 3770 3785 3794 3778 . ± . η c (1 D ) 3799 3809 ψ (2 D ) 4167 4179 ψ (2 D ) 4158 4167 ψ (2 D ) 4159 4142 4153 4191 ± η c (2 D ) 4158 4170 χ (1 F ) 4021 4033 χ (1 F ) 4029 4039 χ (1 F ) 4029 4041 h c (1 F ) 4026 4037 χ (2 F ) 4348 4362 χ (2 F ) 4352 4365 χ (2 F ) 4351 4365 h c (2 F ) 4350 4364 related to each-other in the nonrelativistic expansion anddescribed by the same set of parameters. In an ideal case,only four input states are enough to fit four “arbitrary”parameters of the potential model.Consequently, as a possible test of nonrelativistic ex-pansion, in the present work we consider the “ideal case”and fit the parameters of model according to some partof S-wave charmonia. In such a way we ignore the spin-orbit and tensor interactions during the fitting process.More specifically, we propose a potential model where the parameters are fitted using the four lowest S-wave spin 0and spin 1 states. On top of that we will fit the lowestspin zero 1 S and 2 S states exactly. The parametersof the corresponding interaction potential are also givenin Table I (see the model referred as NR4) and the val-ues of corresponding input states are given in Table II,respectively (see the 4 th column).In order to keep a good accuracy of numerical calcula-tions, during the fitting process we used a basis set with30 to 50 gaussian functions corresponding to the givenvalues of L , S and J . So, the value of the first parameter n max from the three variational parameters is free inputand equals to the definite integer number belonging to theinterval n max ∈ [30 , r and r n max , are foundby minimizing not only the ground state energy E butalso minimizing simultaneously the lowest 10 to 20 radialexcitation energies P n min i = n E n (i.e. n min ∈ [5 , n max ∼
10 with n min ∼ n max ∈ [30 ,
50] and n min ∈ [10 , S -state energy value for NR4 and comparingit with the corresponding experimental value shows thelarge difference, around 45 MeV. Nevertheless, this prob-lem seams to be unavoidable if one fits the parameters tothe whole spectrum, e.g. compare the corresponding NRresult where the difference from the experimental valueis around 33 MeV. However, the next excited 4 S -stateenergy for NR4 is reproduced at almost its averaged ex-perimental value while NR model gives the relatively dif-ferent result. One can conclude that, in general, S-wavestates are reproduced better in NR4 model in comparisonwith NR model.The power of nonrelativistic expansion becomes moreobvious when we include the spin-orbit and tensor inter-actions for the analysis of whole spectrum. While theinput parameters are already fixed we do not need toplay with them anymore. Therefore, the calculations of L ≥ P , 1 P and2 P are reproduced at their experimental value for NR4model. Two states, 1 P and 1 P , among the remain-ing three P-wave states in the table are also reproducedquite well with the differences 3 MeV and 6 MeV from theexperiment, respectively. Only one state 2 P is far fromits experimental value, the difference is 51 MeV. For com-parison, the general fit using NR model reproduces onlyone state 1 P at its experimental value. Another state1 P is same as in the case of NR4 and three (1 P , 1 P and 2 P ) from the remaining four states are reproducedapproximately with 10 MeV differences from the experi-mental values, respectively. The last state 2 P is veryfar from the experimental value and difference 47 MeVis almost same as NR4 case. One could also conclude,that the concentration, respectively, to S-wave (e.g. 1 S and 2 S ) and P-wave (e.g. 1 P and 1 P ) states asan input will lead to more or less similar values of thepotential parameters in comparison with the values inTable II. Summarizing analysis of P-wave states one canconclude that NR4 model much better reproduces theexperimental data in comparison with NR model.It is also interesting to analyze more higher energystates corresponding to NR and NR4 models and com-pare them with the available experimental data. Comingto the D-wave states, one can note that 1 D -state en-ergy value is reproduced relatively better in NR model.However, the situation becomes opposite if we analyze1 D -state energy. Here NR4 model gives relatively bet-ter result in comparison with NR model. Finally, anexperimentally available the highest energy state 2 D isagain reproduced relatively better in NR4 model. Againand in general, D-wave states are also better reproducedin NR4 model.Consequently, one can conclude that, although thenumber of input parameters in NR4 model are chosenin a maximum compact form it gives better result thanthe NR model. From the Table II one can also make ageneral conclusion that the fitting to the S-wave part of the spectrum is completely satisfactory. In a such waywe see that the nonrelativistic expansion in the potentialmodels for describing the fine structure of charmoniumstates indeed works pretty well. IV. SUMMARY
In the present work we aimed at testing the nonrela-tivistic potential model and reparametrization of the non-relatistic potential based on the currently available exper-imental data. In particular, we investigated the appli-cability of nonrelativistic expansion in the potential ap-proaches to the charmonium spectrum. For that purposewe concentrated only on the four lowest S-wave states ofcharmonium spectrum. By doing that we demonstrated,that the concentration to the minimal part of charmo-nium spectrum is enough during the fitting process ofthe values of potential parameters. The model quite sat-isfactorily described the whole spectrum of charmonia.From our studies, one can make a general conclusionthat the nonrelativistic potential approach is indeed goodapproximation in describing the spectrum including thefine-structure of charmonium states.
Acknowledgments
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