P. C. Vinodkumar Department of Physics, Sardar Patel University, V. V. Nagar , INDIA

Transcription

1 Radial and orbital in a Hypercentral Quark Model Department of Applied Physics, S. V. National Institute of Technology, Surat , INDIA Department of Physics, Sardar Patel University, V. V. Nagar-3880, INDIA kaushalphysics@gmail.com P.. Vinodkumar Department of Physics, Sardar Patel University, V. V. Nagar-3880, INDIA p.c.vinodkumar@gmail.com We present using the hypercentral description of the three-body system. The confinement potential is assumed as hyper coulomb plus power potential (hpp ν ) with power index ν. The S, 3S and S radial excited states and first negative parity orbital excited states are computed for different power indices (ν), starting from 0.5 to.0. We also incorporate spin dependent contribution perturbatively. Our calculated results for are in good agreement with known experimental results as well as other theoretical predictions. XV International onference on Hadron Spectroscopy-Hadron 03-8 November 03 Nara, Japan Speaker. c opyright owned by the author(s) under the terms of the reative ommons Attribution-Nonommercial-ShareAlike Licence.

2 . Introduction In beginning of the st century significant progress has been achieved in the experimental facilities at Belle, BaBar, LEO, DF, SELEX, ALIE, LH etc., to study the properties and production of heavy flavour hadrons ]. Out of the huge data bank related to the heavy flavour hadrons, majority of them are in the meson sector. However considerable number of them are also recorded in the baryon sector. So many surprises are also expected in the hadronic sector containing new states coming from the above mentioned world wide experimental setups. Therefore the theoretical calculation of the properties of hadrons containing heavy flavour quarks is desired to conform the experimental observations. At the theoretical front, there exist vast literature on the ground state properties of heavy flavour baryons and few cases of excited states. There are less number of excited heavy baryons available from experimental side due to its fast decay properties. But the recent experimental facilities has recorded and has shown progress in the baryon spectroscopy. The quark composition of Λ + c and Σ + c is udc and Λ + c was the first known charm baryon shortly after the discovery of J/ψ meson. Excited charm baryons were discovered long after the initial observation of the Λ + c ground state. Recently, the DF collaboration ] observed a number of excited charm baryon states including Λ c (595), Λ c (65). The J P quantum numbers for most excited heavy baryons have not been determined experimentally, but are assigned by the Particle Data Group on the basis of quark model predictions. So it is very interesting to assign quantum numbers to excited heavy baryons according to hypercentral quark model. So in this paper, we present radial and orbital in the hypercentral approach which is found to be successful in the study of three body problems.. Methodology The Hamiltonian of the 3-body baryonic systems given in terms of the Jacobi co-ordinates ρ,λ can be expressed in terms of the hypercentral coordinate, 3, ], x as H = P ρ + P λ +V (ρ,λ) = P x +V (x) (.) m ρ m λ m The hyperradial Schödinger equation corresponds to the Hamiltonian given by above Eqn, can be written as d dx + 5 ] d γ(γ + ) x dx x ψ ωγ (x) = me V (x)]ψ ωγ (x) (.) where γ is the hyper angular quantum number and is given by γ = n + l ρ + l λ, n = 0,,...; l ρ and l λ are the angular momenta associated with the ρ and λ variables and ω denotes the number of nodes of the spatial three-quark wave functions. Now for finding solution to the hyperradial Schrödinger equation, we consider the transformation, ϕ ωγ (x) = x 5 ψ ωγ (x). Eqn. (.) then reduces to the form d 5 m dx + + γ(γ + ) mx +V (x) ] ϕ ωγ (x) = Eϕ ωγ (x) (.3)

3 The hyperradial wave function ϕ ωγ (x) is a solution of the reduced Schrödinger equation for the interacting potential defined by V (x) = α s 3 x +βxν. If we compare Eqn. (.3) with the usual three dimensional radial Schrödinger equation, the following correspondance between angular momentum with hyper angular momentum as mentioned in 5] can be made as l(l + ) 5 + γ(γ + ). For computing the mass difference between different degenerate baryonic states, we consider the spin dependent part of the usual one gluon exchange potential (OGEP) given by 6]. Accordingly, the spin-dependent part, V SD (r) contains three types of the interaction terms, such as the spin-spin, the spin-orbit and the tensor part given by 6] V SD (x) = V SS (x) S(S + ) s ρ (s ρ + ) 3 ] ] ) +V ΓS (x)( Γ S +V T (x) S(S + ) 3( S x)( S x) x (.) The hyper spin-orbit term containing V ΓS (x) and the hyper tensor term containing V T (x) describe the fine structure of the baryon states, while the spin-spin term containing V SS (x) proportional to ( s ρ s λ ) = S(S + ) s ρ (s ρ + ) ] 3 gives the spin hyperfine splitting. The spin-orbit term in hypercentral model containing V ΓS (x) proportional to ( Γ S) = J(J + ) Γ(Γ + ) 5 S(S + )]. The ( coefficient ) of these spin-dependent terms of Eqn (.) can be written in terms of the vector V V = xe x Γ(Γ+) and scalar (V S = βx ν ) parts of the static potential as V ΓS (x) = V T (x) = ( 3 dv V m ρ m λ x dx dv ) S dx ( 3 d V V 6m ρ m λ dx ) dv V x dx V SS (x) = (.5) (.6) 3 m ρ m λ V V (.7) The baryon spin average mass is then obtained by the sum of the quark masses plus the binding energy as M W = m i +BE. The ground state and the radial and the orbital excited state masses of i heavy flavour baryons are determined by the sum of spin average masses (M W ) with spin-hyperfine interaction as M B = M W + V SD (x). We fix potential parameter β and hyperfine parameter A for each choices of ν using ground state experimental mass of J P = + and J P = 3 + charm baryons (Figure ). The regularization parameter x 0 is GeV and quark mass parameters are m u = 338 MeV, m d = 350 MeV, and m c = 75 MeV. For radial excited states, potential parameter β is found to vary as β = β 0 ω + γ + 3 and for orbital excited states, potential parameter β is found to vary as β = β 0 ω + γ + where β 0 correspond to the ground state (ω = 0, γ = 0) potential strength. 3. Result and Discussion We have studied the radial and orbital excited states of single heavy charm baryon by solving six dimensional Schrödinger equation numerically with hypercentral potential of the hypercoulomb plus power potential. The computed orbital excited states of Λ + c are in good agreement with experimental known state Λ c (595), Λ c (65) as well as other theoretical prediction at potential index 3

4 Potential Strength in MeV Potential Index Figure : Variation of potential strength β to fix the ground state masses of single heavy charm baryons with respect to potential index ν. Table : Mass of radially excited states including spin interaction for single charm Σ + c baryon with different choices of ν. Potential Mass of Radially Excited states index Present Others Present Others S 0] S 3 0] S S S 3 S S S Table : Mass of first negative parity states including spin, tensor and spin-orbit interaction for Single charm Σ + c baryon with different choices of ν. Baryon Potential present others index ν ] Σ + c ( P ) ] ] ] Σ + c ( P3 ) ] ] ] Σ + c ( P ) ] ] ] Σ + c ( P3 ) ] ] ] Σ + c ( P5 ) ] ] ν =. Experimental known Σ c (79) state is identified as first negative parity orbital excited state of Σ + c with J P =. Same prediction is found in relativistic quark-diquark picture by D. Ebert et al, 0]. We found that the computed excited states are in good agreement with experimental known states at potential index ν =. Thus, the hpp ν model adequately represent the three-body interactions among the quarks constituting the baryons. Acknowledgement: I acknowledge the partial financial support from my parent institute SVNIT Surat as well as IS, INDIA. We specially acknowledge all the supports including lo-

5 Table 3: Mass of radially excited states including spin interaction for single charm Λ + c baryon with different choices of ν. Baryon Potential Present Others Index S 0] Λ + c (S) S Λ + c (S) S Λ + c (3S) S Λ + c (S) cal hospitality from Local Organizing ommittee of HADRON 03. References Table : Mass of first negative parity states including spin, tensor and spin-orbit interaction for Single charm Λ + c baryon with different choices of ν. Baryon Potential Present Others Index ] Λ + c ( P ) ] ] 630 8] ] Λ + c ( P3 ) ] ] 60 8] ] E. Solovieva et al. Belle ollaboration], Phys. Lett. B 67, (009); B. Aubert et al. BABAR ollaboration], Phys. Rev. D 77, 000 (008); J. P. Alexander et al. LEO ollaboration], Phys. Rev. D 8, 0900 (00); T. Aaltonen et al. DF ollaboration], Phys. Rev. D 8, 0003 (0); T. Aaltonen et al. DF ollaboration], Phys. Rev. D 85, 090 (0); A. Ocherashvili et al. SELEX ollaboration], Phys. Lett. B 68, 8 (005). ] E. Santopinto, F. lachello and M. M. Giannini, Eur. Phys. J. A, (998). 3] B Patel, A K Rai and P Vinodkumar, J. Phys. G: Nucl. Part. Phys. 35, (008); Bhavin Patel et al., Pramana- Journal of Physics, 70, (008). ] K. Thakkar, A. Majethiya and P Vinodkumar, hinese Physics, 36(5), (0); K. Thakkar et al., Pramana J. of Phys. 77, (0). 5] R Bijker et al., J. Phys. A : Math. Gen (998). 6] Voloshin M B, Prog. Part. Nucl. Phys. 6, 55 (008); arxiv:hep-ph/07.556v3. 7] W. Roberts and M. Pervin, Int. J.Mod. Phys. A 3, 87 (008). 8] S. apstick and N. Isgur, Phys. Rev. D 3, 809 (986). 9] J. Beringer et al. (Particle Data Group), Phys. Rev. D (0). 0] D. Ebert et al., Phys. Lett. B 659, 6 (008); D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 8, 005 (0). 5