Heavy quarkonium properties from Cornell potential using variational method and supersymmetric quantum mechanics

Transcription

1 Pamana J. Phys. (06) 87: 73 DOI 07/s c Indian Academy of Sciences Heavy quakonium popeties fom Conell potential using vaiational method and supesymmetic quantum mechanics ALFREDO VEGA and JORGE FLORES Instituto de Física y Astonomía, Univesidad de Valpaaíso, Avenida Gan Betaña, Valpaaíso, Chile Coesponding autho. alfedo.vega@uv.cl MS eceived Octobe 05; evised 0 Novembe 05; accepted 5 Januay 06; published online 8 Octobe 06 Abstact. Using the vaiational method and supesymmetic quantum mechanics we calculated, in an appoximate way, the eigenvalues, eigenfunctions and wave functions at the oigin of the Conell potential. We compaed esults with numeical solutions fo heavy quakonia c c, b b and b c. Keywods. PACS Nos Quakonium; vaiational method; supesymmetic quantum mechanics. 4.0.Lq; w; 4.40.Pq;.80.Fv. Intoduction Since the discovey of J/ψ in 974 [,], the study of heavy quakonia has been vey valuable in hadon physics because they involve the non-petubative aspect of quantum chomodynamics (QCD) and thee ae many expeimental data about them [3 5]. Fom a theoetical pespective, heavy quakonia have been studied using seveal models [6,7], with the non-elativistic potential models standing out due to thei simplicity, whee quak inteaction is modelled using potential enegy in the usual Schödinge equation. This potential pictue can be justified fo heavy quakonia by using non-elativistic QCD (NRQCD) with a potential that can be expessed as a one-ove quak mass expansion, whee the Conell potential is a good fist tem [8 ]. The Conell potential [,3] was one of the fist potentials poposed to descibe the inteaction between heavy quaks. It coesponds to a Coulomb potential plus a linea confinement pat. Theefoe, the Conell potential consides geneal popeties of quak inteactions as asymptotic feedom and confinement. Although thee is a lage amount of liteatue elated to quak potentials, we would like to list efs [7, 9]. Although this list is incomplete, these efeences ae a good stating point on this topic. The Schödinge equation with the Conell potential does not povide exact analytical solutions. Although it can be solved using numeical methods [0,], it is inteesting to obtain analytical solutions (at least appoximate ones) that offe the possibility of diffeent applications. Fo example, in many hadon physics applications in light-cone fame, it is common to see the use of an analytical ansatz inspied by hamonic oscillato wave function in constant time fame in ode to achieve the light font wave function used in calculations (fo example, see [ 4]). We believe that it could be inteesting to stat fom a moe ealistic wave function, fo example, associated with the Conell potential. In this wok, we solved, in an appoximate way, the Schödinge equation with the Conell potential using a pocedue that coesponds to an adaptation of the method suggested in [5,6], which consideed the usual vaiational methods with supesymmetic quantum mechanics (SUSY QM). Some additional examples using SUSY QM and the vaiational method can be found in [7 9]. SUSY QM [30] was bon at the beginning of the 980s in studies about beaking supesymmety in theoies of quantum fields with exta dimensions [3], and is a technique that allows us to get isospectal potentials fo the Schödinge equation. The isospectal potential coesponds to the supesymmetic patne potential of the oiginal, with the paticulaity that the gound state of the oiginal potential is not pesent in the spectum of the associated isospectal potential. Theefoe,

2 73 Page of 7 Pamana J. Phys. (06) 87: 73 the gound state of the supesymmetic patne potential is elated to the fist excited state of the oiginal potential. This pocedue can be epeated to get successive potentials, the gound states of which ae elated by some standad tansfomations in SUSY QM to the diffeent states of the oiginal potential; hence, SUSY QM can be used to build an infinite family of isospectal potentials. Thus, if we use the vaiational method to get solutions fo the gound state of diffeent supesymmetic patnes of the Conell potential, we can obtain the spectum and wave functions fo heavy quakonium. It is impotant to mention that the standad vaiational method has been used to study heavy quakonium popeties by consideing diffeent phenomenological quak potentials [3,33]. The pocedue descibed in the pevious paagaph was used in this wok to get appoximations to eigenvalues and eigenfunctions fo the Schödinge equation with the Conell potential, and we used it to study heavy quakonia c c, b b and b c. Additionally, we paid special attention to the wave function at the oigin (WFO), which is an impotant quantity involved in seveal decay ates of heavy quakonium. Although the pocedue descibed in this pape can be used in geneal to study adial and angula excitations, we esticted ou study to only to S states. This pape consists of fou sections. In we summaize the main elements of SUSY QM used in this wok. Section 3 is dedicated to obtaining appoximate calculations of enegies, wave functions and WFO fo heavy quakonium using the vaiational method and SUSY QM, and in 4 we discuss ou esults and conclusions.. Basics of SUSY QM In this section we summaize the main elements of SUSY QM used in the following sections to calculate heavy meson popeties using the Conell potential. Fo moe details, we suggest ef. [30]. We conside the Schödinge equation fo the gound state with an eigenvalue equal to zeo (this can be done without losing geneality, because the potential can be edefined by adding a constant tem equal to minus gound-state enegy). Thus, the wave function ψ 0 obeys H ψ 0 (x) = h Then, V (x) = h m m d ψ 0 (x) dx + V (x)ψ 0 (x) = 0. () ψ 0 (x) ψ 0 (x). () The Hamiltonian H can be factoized as H = A A, whee A = h m d dx +W(x) and A = h m d dx +W(x). With this, we obseved that fo a known V, the supepotential W satisfies the Riccati equation V (x) = h dw(x) + W (x). m dx The solution fo W(x) in tems of the gound-state wave function is W(x) = h ψ 0 (x) m ψ 0 (x). (3) Additionally, with opeatos A and A it is possible to build a new Hamiltonian H given by H = AA, and this new Hamiltonian can be expessed as H = h d m dx + V (x), whee V (x) = h dw(x) + W (x). m dx Potentials V (x) and V (x) ae known as supesymmetic patne potentials, and they have seveal inteesting popeties (see [30]). It is impotant to point out that eigenvalues and eigenfunctions of H and H ae elated to E n () = E () n+ ; E() 0 = 0 (4) ψ () n = and E () n+ ψ () n+ = E () n Aψ () n+ (5) A ψ () n. (6) We pay special attention to the elationship in the spectum of H and H because, with the exception of the gound state of H (that did not appea in H ), additional levels ae the same in both Hamiltonians, i.e., potentials V and V ae isospectals except fo E () 0. Similaly, stating fom H and its gound state, we can build Hamiltonian H 3, isospectal to H (in the same sense that H and H ae isospectals), and if we

3 Pamana J. Phys. (06) 87: 73 Page 3 of 7 73 Although the pocedue we consideed can be used to study popeties of diffeent adial and angula excitations, in this wok we only conside the case l = 0 (S states). Thus, the effective potential is equal to the Conell potential, V eff () = V()= κ + β, (8) Figue. Schematic epesentation of potential V and its fist two supesymmetic patnes V and V 3 with its coesponding spectum. The gound state in V is not pesent in V, gound state of V is not pesent in V 3 andsofoth.the shape of each potential is diffeent, but this issue is disegaded to explain that the gound state of one potential is not pesent in its supesymmetic patne. epeat this pocedue, it is possible to obtain a family of isospectal potentials whee, as can be seen in figue, the gound state of V is elated to the fist excited level of V, the gound state of V 3 is elated to the fist excited level of V and the second level of V, and so foth in ode to achieve diffeent levels of the oiginal potential V. Accoding to the pevious paagaph, the gound state of H coesponds to the fist excited level of H. This fact is especially impotant because the common vaiational method in quantum mechanics is a good tool to obtain appoximate values fo the gound states in the Schödinge equation. Theefoe, we can use this simple pocedue to get appoximate solutions fo gound states fo diffeent supesymmetic patne potentials by using the vaiational method, and we can use SUSY QM elationships to get appoximate solutions fo diffeent levels of a potential of inteest V, such as the Conell potential. 3. Solutions fo the Conell potential with vaiational methods and SUSY QM In this section we use the pocedue descibed in. This was suggested in [5,6] and we adapted it to calculate enegies, wave functions and WFO fo S states in quakonia c c, b b and b c using the Conell potential. Conside U nl () = R nl (), which satisfies d U nl () μ d + V eff ()U nl () = E n U nl (), (7) whee h =, μ is the educed mass fo heavy quakonium consideed, and V eff () = V()+ l(l + ) μ. whee the paametes involved ae [,3,34,35] κ = 0.5; β = (.34) GeV and the quak masses ae [ ] [ ] GeV GeV μ c =.84 and μ b = 5.8. c With this potential it is possible to gain a mass spectum fo quakonia by using m n ( Q Q ) = m Q + m Q + E n +, (9) whee m Q and m Q ae masses of the quaks and the antiquaks inside the quakonia consideed, E n is the eigenvalue associated with (8), and is a constant that must be added to the Conell potential in (8). Fo the vaiational method we conside the tial wave function U() = N γ e ab. (0) This tial wave function can be used to obtain the gound state in the Conell potential and in its supesymmetic patnes to calculate the successive levels in Conell potential. In this wave function, γ takes the values,, 3,... depending on the calculation of the gound state of potentials V, V,... (accoding to ). Changing this paamete in this way is impotant, because to gain appoximations fo the wave functions of the Conell potential fo diffeent levels, it will be necessay to apply successive tansfomations defined by (6), and this choice tuns out to be the only possibility to gain finite WFOs with a tial wave function like (0). Paametes a andb ae vaiational paametes and N is the nomalization constant given by N = (a) (+γ)/b b Ɣ (( + γ)/b). To calculate the gound state fo the Conell potential we use (7) and the tial wave function with γ =. Thus, the expectation value of the enegy is E = U() d μ 0 d U()d + ( κ ) + β U ()d. 0 c

4 73 Page 4 of 7 Pamana J. Phys. (06) 87: 73 By using U() given in (0), we get an expectation value of the enegy that depends on paametes a and b (E(a,b)), and by minimizing this, an appoximate value fo the enegy and values fo the paametes a and b ae obtained. In this case, we can call the paametes associated with the gound state a 0 and b 0. Then, WFO is calculated fo the gound state. In this case we conside two appoaches that ae equivalent when woking with the exact solutions, but they have diffeent values when using appoximate wave functions. In this case we use R() = U()/. The fist appoach consideed in WFO calculations, and hencefoth called Method, is based on a wellknown expession (valid fo S states only) that elates WFO to expectation values fo the fist deivative of the potential dv() (0) = μ π d Fo S states () = R(). 4π. Theefoe, dv() R(0) = μ. d The second appoach is simply to take 0inthe wave function. This is what we have called Method in the following. Accoding to this, WFO fo a gound state can be found diectly fom R() = (U()/) and (0) (with γ = fo the gound state), and we get R(0) = N = (a 0) (+γ)/b 0 b 0 Ɣ (( + γ )/b 0 ). Once we finish with the gound state, we calculate the fist adial excitation in the Conell potential. Note that using vaiational method to appoximate the excited states is not a simple task because it is impotant to ensue that the tial eigenfunctions ae othogonal. In this pape, we solve the poblem of the fist adial excitation by using the pocedue discussed in. Then, we explain how to use SUSY QM and the vaiational method to solve S states. Peviously, eq. (0) was used with γ = togetsolutions fo the gound state, thus using the common vaiational method, enegy values fo the S states ae obtained, and we fit the paamete in the wave functions. Then 0 index in the vaiational paametes indicates that these paametes ae associated with the gound state. With this tial wave function (elated to the gound state of the Conell potential) we obtain the supepotential W () = U () μ U() = + a 0b 0 b0. μ The index in W indicates that stating fom solutions of potential V (Conell in this pape), we can build a potential V (an appoximate supesymmetic patne fo the Conell potential) V () = [W ()] + ( ) dw () μ d ( + a0 b 0 b 0( 3 + a 0 b 0 + a 0 b 0 b ) 0) V () = μ. Next, the vaiational method is used to get an appoximate value fo the gound-state enegy of potential V. In this case, the tial wave function has the shape (0) with γ =, and the enegy is elated to the fist excited state of V. Fo V, the enegy expectation value depends on a 0, b 0 (fixed in pevious steps when we calculate the gound state of V )anda and b, which must be fixed once this expectation value has been minimized. It is impotant to note that in ou discussion fo SUSY QM we conside a gound state with eigenvalue equal to zeo; theefoe, the gound state of enegy fo V epesents E, and consequently the enegy fo the fist excited state is E = E 0 + E. Theeafte, the second excited level in the Conell potential is obtained. As we have an appoximate solution fo V, we can build W 3 and obtain its supesymmetic patne V 3. If we find the gound state of this new potential using the vaiational method using (0) with γ = 3, we can find the enegy of the second excited state of the oiginal potential by using E = E + E 3, and so foth. Table shows the enegy values calculated with the method used in this pape, and we compae it with an exact numeical solution obtained using a MATHEMATICA pogam called mathschoe.nb [0]. Hee we would like to show how we obtained the wave function fo the excited levels in the Conell potential. If we have a solution fo the gound state of the potential V, which we call fo example ψ () 0, additionally as we know W (we used it to build V ),

5 Pamana J. Phys. (06) 87: 73 Page 5 of 7 73 Table. Enegy values (in GeV) fo heavy quakonia c c, b b and b c. Column Exact shows the solution that coesponds to numeical calculations using mathschoe.nb with step h = 000 and the column Ous shows the enegies calculated in this wok. c c b b b c E n Exact Ous Exact Ous Exact Ous s s s it is possible to get a wave function fo the fist excited state of V using the equation ψ () A ψ() 0. In pinciple, fo exact nomalized solutions, (6) gives the ight nomalization, but as we ae woking with appoximate solutions we pefe to nomalize each wave function at the end, and theefoe in pevious expessions we used symbol. Opeato A tansfoms the solution ψ() 0 fo the gound state of V in a solution fo the fist excited state of V, and thus the wave function of the fist excited state of V is obtained. ψ () ( μ + W () ) ψ () 0. In a simila way, we can build the wave function fo the second excited state of the Conell potential stating fom the gound state of V 3 ψ () A A 3 ψ(3) 0 ( )( ) ψ () + W () + W 3 () ψ (3) μ μ 0. Figue shows the adial density of pobabilities calculated with the method discussed hee that joins SUSY QM and vaiational methods, and the esults ae compaed with the esult obtained numeically using mathschoe.nb. The wave functions ae used to calculate WFO, and tables, 3 and 4 show a summay of ou esults obtained by using Method and Method Figue. Density functions of adial pobability. The continuous line coesponds to the numeical and the dashed line coesponds to the method used in this pape. The fist column coesponds to the gound states, the second to the fist excited state and the thid to the second excited state. The uppe ow coesponds to c c, the middle to b b and the lowe ow to b c.

6 73 Page 6 of 7 Pamana J. Phys. (06) 87: 73 Table. Mass spectum (in GeV) fo heavy quakonia c c, b b and b c. Columns Exact and Ous epesent the same as those in table, and the column called Exp coesponds to expeimental values accoding to [36]. c c b b b c m Q Q Exp. Exact Ous Exp. Exact Ous Exp. Exact Ous s s s Table 3. Compaison of WFO of the fist thee enegy values in heavy quakonia c c, b b and b c. The values calculated with the numeical solutions ae given in the column Method(abbeviatedbyM). c c b b b c R(0) Exact M Exact M Exact M s s s Table 4. Compaison of WFO of the fist thee enegy values in heavy quakonia c c, b b and b c. The values calculated with the numeical solution ae given in column Method(abbeviatedbyM). WFO c c b b b c R(0) Exact M Exact M Exact M s s s Conclusion and discussion We used a pocedue to solve, in an appoximate way, the Schödinge equation with the Conell potential using a vaiational method and SUSY QM. This is phenomenologically inteesting, because the Conell potential can descibe some popeties of heavy quakonium, fo example its masses, as shown in table. Theefoe, it is useful to have analytical wave functions, as the ones povided in this pape, which can be useful in additional phenomenological applications in mesonic physics. The esults in table show the values fo the enegies of the fist thee excited states, which ae close to the exact computation, especially fo the gound and the fist excited states. The same happens with the wave functions shown in figue, whee the numeical and appoximate wave functions ae almost the same fo the gound state and vey close to the fist excited state, but when we conside highe adial excitations both wave functions ae diffeent. We also calculated WFO. To do so, we consideed two methods that ae equivalent when woking with exact wave functions, but as shown in tables and 3, we obtained diffeent esults when using appoximate wave functions. Method, based on calculations of the expectation values of the fist deivative of the potential, deliveed a bette esult than Method, which was based on putting 0 diectly. Consi deing that R() = U()/,asU( 0) 0, to use 0 diectly could cause poblems as we had an expession 0/0, and with this it was possible to undestand the diffeence in R(0) by using Methods o. This shows that the best choice to evaluate

7 Pamana J. Phys. (06) 87: 73 Page 7 of 7 73 WFO is Method if thee is an appoximate wave function. Ou esults suggest that the appoximate method discussed can poduce esults in ageement with numeical exact solutions fo lowe states, and in disageement with highe excited levels. This is not supising, because we solved the Schödinge equation in an appoximate way (we used the vaiational method) with an appoximate potential (we obtained supesymmetic patne potential stating fom vaiational tial functions). Theefoe, even if we stat with a good tial wave function of the gound state, fo highe adial excitations the method begins to poduce esults which ae inconsistent with the numeical solutions. Howeve, fo lowe states this appoach povides good esults. Undoubtedly, if we use tial wave functions with seveal paametes we can impove ou esults, but as we have shown, this method woks well fo lowe states, and it could be helpful to use it as a complement with a method that woks fo highe excitations, such as WKB. This appoach gives us analytical expessions fo wave functions close to numeical solutions and they ae othogonal. So this can be used to calculate othe heavy quakonium popeties. Finally, we would like to etun to what we mentioned at the beginning of this pape. In many applications of hadonic physics in light-cone fame, it is common to see the use of an analytical ansatz inspied in hamonic oscillato wave function in a constant time fame, and this is utilized to get the light font wave function used in calculations (fo examples, see [ 4]). We believe this could be an inteesting stat of a moe ealistic wave function, fo example, associated with the Conell potential as a method that joins SUSY QM and the vaiational method discussed in this aticle. Acknowledgements This wok was suppoted by FONDECYT (Chile) unde Gant No. 480 and by CONICYT (Chile) unde Gant No The authos wish to thank DGIP of the Univesidad de Valpaaíso and Maía Angélica Rojas fo thei collaboation in the pepaation of this manuscipt. Refeences [] E598 Collaboation: J J Aubet et al, Phys. Rev. Lett. 33, 404 (974) [] SLAC-SP-07 Collaboation: J E Augustin et al, Phys. Rev. Lett. 33, 406 (974) [3] M B Voloshin, Pog. Pat. Nucl. Phys. 6, 455 (008), axiv: [hep-ph] [4] C Patignani, T K Pedla and J L Rosne, Ann. Rev. Nucl. Pat. Sci. 63, (03), axiv:.655 [5] N Bambilla, S Eidelman, B K Heltsley, R Vogt, G T Bodwin, E Eichten, A D Fawley, A B Meye et al, Eu. Phys. J. C 7, 534 (0), axiv: [hep-ph] [6] B Ginstein, Int. J. Mod. Phys. A 5, 46 (000), hepph/9864 [7] W Lucha, F F Schobel and D Gomes, Phys. Rep. 00, 7 (99) [8] N Bambilla, A Pineda, J Soto and A Vaio, Phys. Rev. D63, 0403 (00), hep-ph/00050 [9] N Bambilla, A Pineda, J Soto and A Vaio, Nucl. Phys. B 566, 75 (000), hep-ph/ [0] G T Bodwin, D Kang and J Lee, Phys. Rev. D74, 0404 (006), hep-ph/ [] J L Domenech-Gaet and M A Sanchis-Lozano, Phys. Lett. B 669, 5 (008), axiv: [hep-ph] [] E Eichten, K Gottfied, T Kinoshita, K D Lane and T M Yan, Phys.Rev. D7, 3090 (978), Eatum, ibid. D, 33 (980) [3] E Eichten, K Gottfied, T Kinoshita, K D Lane and T M Yan, Phys. Rev. D, 03 (980) [4] J L Richadson, Phys. Lett. B 8, 7 (979) [5] W Buchmulle and S H H Tye, Phys. Rev.D4, 3 (98) [6] S N Gupta and S F Radfod, Phys. Rev.D4, 309 (98) [7] S N Gupta, S F Radfod and W W Repko, Phys. Rev. D34, 0 (986) [8] P Gonzalez, A Valcace, H Gacilazo and J Vijande, Phys. Rev.D68, (003), hep-ph/ [9] C O Dib and N Neill, Phys. Rev. D86, 0940 (0), axiv:08.86 [hep-ph] [0] W Lucha and F F Schobel, Int. J. Mod. Phys. C 0, 607 (999), hep-ph/98453 [] J L Domenech-Gaet and M A Sanchis-Lozano, Comput. Phys. Commun. 80, 768 (009), axiv: [hep-ph] [] T Huang, B Q Ma and Q X Shen, Phys. Rev. D49, 490 (994), hep-ph/94085 [3] T Peng and B Q Ma, Eu. Phys. J. A 48, 66 (0), axiv: [hep-ph] [4] T Wang, D X Zhang, B Q Ma and T Liu, Eu. Phys. J. C 7, 758 (0), axiv: [hep-ph] [5] E Gozzi, M Reute and W D Thacke, Phys. Lett. A 83, 9 (993) [6] F Coope, J Dawson and H Shepad, Phys. Lett. A 87, 40 (994) [7] E Digo Filho and R M Ricotta, Mod. Phys. Lett. A 0, 63 (995), hep-th/95073 [8] E Digo Filho and R M Ricotta, Phys. Lett. A 69, 69 (000), hep-th/99054 [9] G R Peglow Boges and E Digo Filho, Int. J. Mod. Phys. A 6, 440 (00) [30] F Coope, A Khae and U Sukhatme, Phys. Rep. 5, 67 (995), hep-th/ [3] E Witten, Nucl. Phys. B 88, 53 (98) [3] Y B Ding, X Q Li and P N Shen, Phys. Rev. D60, (999), hep-ph/ [33] G R Booun and H Abdolmalki, Phys. Sc. 80, (009) [34] E J Eichten and C Quigg, Phys. Rev. D5, 76 (995), hep-ph/ [35] E J Eichten and C Quigg, Phys. Rev. D49, 5845 (994), hep-ph/9400 [36] Paticle Data Goup Collaboation: K A Olive et al, Chin. Phys. C 38, (04)