Regge Trajectories Analysis to D SJ(2317) ±, D SJ (2460) ± and D SJ (2632) + Mesons

Transcription

1 Regge Trajectories Analysis to D SJ(37) ±, D SJ (460) ± and D SJ (63) + Mesons arxiv:hep-ph/04084v 0 Aug 004 Ailin Zhang Department of Physics, Shanghai University, Shanghai, 00436, China Abstract The properties of the new observed charmed strange mesons D SJ (37)±, D SJ (460) ± and D SJ (63) + are simply listed. The status of investigations of these states is reviewed. An analysis to these states with Regge trajectories was made. We found that D SJ (37)± and D SJ (460) ± could be arranged as (0 +, + ), but D SJ (63) + seems not possible to be an orbital excited tensor particle. As a byproduct, the non-strange charmed mesons including D (47) and D (637) + were analyzed also. PACS numbers:.39.-x, 3.0.Gd, 3.5.Gv, 4.40.Gx

2 Introduction The problem of the Quantum Chromodynamics(QCD) spectrum is the central issue in the nonperturbative QCD and is connected to problems of confinement and mass generation in QCD. Charmed strange mesons are an important system to study hadrons spectrum for their internal heavy and light quark(anti-quark). There were limited experimental data for this system before, but the situation changed a lot since last year. Experiments have opened a fascinating period about charmed strange mesons. Several new states have been observed. DSJ (37)± was first observed in D S + π by BaBar[] at SLAC, then confirmed by CLEO[] at CESR, Belle[3] at KEK and Focus[4] at Fermilab. Its mass is 37.4 ± 0.9 MeV from PDG[5], about 40 MeV below DK threshold. The full width Γ < 4.6 MeV at 90% confidence level. In the confirmation of DSJ (37)±, D SJ (460) ± was first reported by CLEO[] in DS π0 final states, and later was observed by BELLE[6] and BaBar[7]. This state has mass ±.3 MeV[5], about 50 MeV below D K threshold, and full width Γ < 5.5 MeV at 90% confidence level. Very recently, a new surprisingly narrow charmed strange meson, D SJ (63) +, was observed by SELEX[8] at Fermilab in the decay channels D S + η and D 0 K +. The mass is found to be 63.6 ±.6 MeV, about 74 MeV and 6 MeV above D 0 K + and D S η threshold, respectively. The width Γ < 7 MeV at 90% confidence level. The relative branching ratio for the two observed modes Γ(D 0 K + )/Γ(D S + η) = 0.6 ± Its decay property is special. These experimental data has triggered enormous discussions about the nature of these states. Considering the unclear dynamics of their decays, we will put aside the issue on the decay modes(narrow width) of these states and pay attention to their spectrum alone. In a unified quark model, the spectrum of charmed strange c s mesons has been computed by Godfrey-Isgur-Kokoski[9] with a universal one-gluon-exchange-plus-linear-confinement potential. The predicted masses are higher than the observed experimental data. In the frame of relativistic quark model, the mass spectrum of orbitally and radially excited D and B mesons was calculated[0] in the m Q limit. The /m Q corrections was carried out also. The predicted masses are lower than the observed experimental data. It is usually believed that lattice theory may give a final solution to QCD. The spectrum of S-wave and P-wave heavy light mesons has been computed with NRQCD on the lattice in the quenched approximation[]. The spectrum of radially and orbitally excited heavy light states are calculated in literature[]. However, as we can realize, the lattice results in these calculations are higher compared with current experimental results. The identification of these new observed mesons will pose a serious challenge for theoretical calculations in normal quark potential model and lattice. For heavy light system, it has a special role in hadron spectrum. In Heavy Quark Effective Theory (HQET)[3], the spin of the heavy quark decouples from the rest of the system and the all meson properties are determined by light degrees of freedom alone. It is then possible to classify the heavy light hadrons using the light degrees of freedom of spin j as a good quantum number. Heavy light mesons can be collected in doublets with a peculiar spin and parity j P. Members of each doublet degenerate in mass in leading order. So the spin-parity in ground states is j P =. This doublet comprises two states with total spin-parity J P = (0, ), which

3 correspond to S 0 and 3 S states in normal quark models. The first excited states involving a P-wave excitation have light degrees of freedom j P = + or 3+. The two doublets have J P = (0 +, + ) and J P = ( +, + ), respectively. When the spontaneous breaking of chiral symmetry was taken into account, the heavy light system can be analyzed in the chiral quark model[4]. In this model, the mesons are predicted to appear in parity-doubled bound states, which transform as linear representations of the light quark chiral symmetry. Then it is natural to conclude that parity doublet has the same mass splittings. For the low lying mesons, the chiral partner of the (0, ) ground state is (0 +, + ). In the framework of chiral quark model, the spectrum of excited D mesons was calculated by W. Bardeen et al[5] recently. The obtained results are in good agreement with experimental data. The calculations of the spectrum of heavy light mesons with other methods such as QCD string[6], MIT bag model and QCD sum rules[7] will not be introduced here. Before the analysis with Regge trajectories, we will give a simple review to the explanations about the nature of these new observed states in literatures. After analysis of its special decay properties and the spectrum calculations based on Godfrey- Isgur-Kokoski and lattice, DSJ(37) ± was explained as DK meson molecule and Dπ atom in literatures[8]. This state was proposed to be a four quark state in [9]. In literatures[5, 7, 0], it was considered as the P-wave 3 P 0 c s mesons. There were baryonium explanation [] and mixed state explanation[] also. The case is similar for D SJ (460) ± except for the P-wave P c s explanation. Based on the fact that the decay mode D 0 K + is anomalously suppressed with respect to D S + η, D SJ (63) + was suggested to be a four quark state with configuration [cd][ d s[3]. A four quark state explanation was obtained in [4] also. There were tetraquarks explanations to this state[6]. In the normal quark model, this state was explained as the first radial excitation state of DS() ± [5]. For a systematic review to this excited subject, literatures[7] are recommended. Till now, all calculations of hadron spectrum have relied on models. In this paper, we will make a phenomenological analysis to these states using the linear structure of Regge trajectories. In fact, when the new data about these resonances has been collected, it is possible to study the charmed spectrum with Regge trajectories. Regge trajectories Several decades ago, it was known from meson phenomenology that the square of the hadron masses depend approximately linearly on the spin of the hadrons. A Regge trajectory is a line in a Chew-Frautschi[8] plot representing the spin of the lightest particles of that spin versus their mass square, t: α(t) = α(0) + α t () where intercept α(0), and slope α depend on weekly on the flavor content of the state lying on the corresponding trajectory. 3

4 For light mesons, experimental data and some models give α 0.9 GeV. () For q q mesons, based on much trial and experimentation, the flavor dependence was assumed to be on the quark mass combination m + m of the component. A global description to the Regge trajectories for all flavors was constructed as[9] α(m + m, t) = α I (m + m, 0) + α (m + m )t, (3) where the subscript I refers to the leading trajectory. When we limit ourselves to those mesons for which the lowest physical state is at J =, we have and α I (m + m, 0) = 0.57 (m + m ), (4) GeV α (m + m ) = 0.9 GeV [ + 0.( m +m GeV )3/ ]. (5) For leading trajectories whose ground states begin at J = 0, they have an intercept approximately 0.5 MeV lower from the leading trajectory and follow a similar pattern. Eq. [3] was constructed from a comprehensive phenomenological analysis of available experimental data for mesonic resonances of light, medium and heavy flavors. It has been supplemented by results from various phenomenological models. For radial excited light q q mesons, the trajectories on (n, M )-plots are obtained by[30] M = M 0 + (n )µ, (6) where M 0 is the mass of basic meson, n is the radial quantum number, and µ is the slope parameter of the trajectory: µ is approximately the same for all trajectories. In addition to mesons, baryons[3], glueballs[3] and hybrids[33] have been studied with Regge trajectories also. Regge trajectories was put forth at first as a good working hypothesis, and then verified by large amount accumulated experimental data. After comprehensive study on experimental data, people gradually realize that Regge trajectories may deviate from straight line[34], depending on peculiar family of mesons, baryons, glueball or hybrids and energy region. However, for mesons with small J, the Regge trajectories are linear. The detailed behavior of Regge trajectories has been studied by more fundamental theories[35]. 3 Analysis of Charmed Family After simple knowledge to Regge trajectories, we start our analysis to charmed strange mesons. In quark model, q q mesons could be marked by their quantum numbers, In S+ L J. Their parity 4

5 States J PC n S+ L J j p PDG note D S (969) ± 0 + S 0 DS ()± 3 S J P =?? consistent with DSJ (37)± P + 0 J, P need confirmation D SJ (460) ± + + P D S (536) ± P + J, P need confirmation D S (573) ± P + J P =?? consistent with + D SJ (63) + 3 S J P =?? Table : Spectrum of Charmed and Strange Mesons. and charge conjugation are P = ( ) L+, C = ( ) L+S, respectively. From PDG[5], we get Table. In this table, the entries in the first volume are those observed mesons, the entries in the last volume are from PDG, while the entries in J PC, n S+ L J and j P (light degree) for those unconfirmed mesons in PDG are favored assignment by theoretical analyses. In chiral quark model, the new observed D SJ (37)±, D SJ (460) ± are suggested to be the (0 +, + ) states, chiral doubler of the observed (0, ) states: D S (969) ± and D S() ±. These states really have the similar splitings D SJ (460) ± D S() ± D SJ(37) ± D S (969) ± 348 MeV. (7) Following this suggestion, based on Table, we proceed the analysis with Regge trajectories. As well known, the, ++ determines a trajectory, and 0 +, + determines another trajectory. The first trajectory is believed to be D S() ± ( ), D S (573) ± ( ++ ). From Eq. [3], the slope of the trajectory could be easily obtained α (m c + m s ) = The second trajectory is assigned as D S (969) ± (0 + ), D SJ (460) ± ( + ), where the slope of the trajectory α (m c + m s ) =.573. GeV GeV. (8) GeV GeV. (9) Obviously, the slope of the two trajectories for D S mesons is almost the same, which is the natural conclusion of Eq. [3]. Our simple analysis supports the assignment for these mesons: D S (969) ± (0 + ), D S ()± ( ), D SJ (460) ± ( + ), D S (573) ± ( ++ ). Especially, there exists no phenomenon called as spin-orbit inversion[0, 36], which may have relation with the dynamics spin-dependence of the confinement. Through the analysis to its decay final states, D SJ (63) + should have J P = 0 +,, +,.... This state was suggested[5] as the first radial excited state of D S ()± ( ), but its narrow 5

6 width brings confusion. It is believed that orbitally excited states are narrower than the radially excited ones. If we assign the new observed D SJ (63) + as the orbitally excited ++ state, then we have the trajectory D S ()± ( ), D SJ (63) + ( ++ ), where the slope α (m c + m s ) =.63. GeV GeV. (0) The slope is much smaller than previous GeV, and the two Regge trajectories will not be parallel. Therefore, the possibility that D SJ (63) + is a ++ tensor resonance is excluded. As another application of Regge trajectories, we now make an analysis to the non-strange charmed mesons. The observed non-strange charmed states are collected in Table. The, ++ trajectory is regarded as D (00) ± ( ), D (460) ± ( ++ ), where the slope α (m c + m u,d ) = GeV GeV. () The obtained results α (m c + m s ) and α (m c + m u,d ) are consistent with the flavor dependence of Eq. [5]. The D(869) ± has 0 +, but the + is missing! Recently, the new observed D (308) 0 and D (47)[37] were suggested as the (0+, + ) chiral doubler of the (0, ) states: D(869) ± and D (00) ± [7]. If we fit D (47) as the missing + state, then we have trajectory D(869) ± (0 + ), D (47) (+ ). The slope of this trajectory α (m c + m u,d ) = GeV 0.47 GeV, () which is much smaller than previous GeV. Obviously, this assignment of D (47) is inconsistent with Regge trajectory. Especially, the linearity of the Regge trajectories gives a mass prediction to the missing + state, 350 MeV. Similar to D SJ (63) +, the recently observed D (637) + by DELPHI in the D ππ channel[38] is impossible to be assigned as a tensor state. Suggestions about this state could be found in literatures[39]. If D (637) +, D SJ (63) + are really the first radial excited states of D (00) ± ( ), DS() ± ( ), their spectrum are exotic. The mass of non-strange charmed meson lies below corresponding charmed strange meson, but their first radial excited states have an inversion relation. If this the case, their trajectories on (n, M )-plots deviate from Eq. [6] for light mesons. 6

7 States J PC n S+ L J j p PDG note D(869) ± 0 + S 0 D (00) ± 3 S +? +? J, P need confirmation D0 (308)± P 0 D (47) + P D (40) P J, P need confirmation D (460) ± P J P = + assignment strongly favored D (637) + 3 S J P =?? Table : Spectrum of Non-strange Charmed Mesons. 4 Conclusions and Discussions In this paper, properties of the new observed charmed strange mesons were listed and some main investigations to these states were introduced. Considering the rapid accumulations of literatures, this introduction is not comprehensive. The complicated decay properties have not been touched, only their spectrum is concerned. From Regge trajectories, we made an analysis to the charmed strange and non-strange mesons. It is really interesting that some conclusions could be made through simple algebra with this tool. Our conclusions could be listed:, The assignment of D SJ (460) ± to + is reasonable., The assignment of D SJ (63) + as the ++ state could be excluded by Regge trajectories. 3, D (47) seems not a + state, the mass of the right candidate of + non-strange charmed state is predicted, 350 MeV. 4, The assignment of D (637) + as the ++ state could be excluded. 5, If D (637) +, D SJ (63) + are really the first radial excited states of D (00) ± ( ), D S() ± ( ), their spectrum are exotic and their Regge behavior is different from that in light mesons system. Obviously, both their spectrum and decay features are important. The discoveries of these new states challenge our existed theories. However, when we turn around to look at the entries in Table and, we find that we still have little knowledge to heavy light charmed mesons. Quantum numbers of some states have not been measured, quantum numbers of some states need confirmation. Some predicted neighboring slightly more massive states should be searched. More decays modes should be detected and analyzed. No matter what is going on, the identification of these new states will definitely challenge our theories and will push our theories ahead. Acknowledgement This work is supported by National Natural Science Foundation of China. 7

8 References [] B. Aubert, et al., BaBar collaboration, Phys. Rev. Lett. 90, 400(003), hep-ex/ [] D. Besson, et al., CLEO collaboration, Phys. Rev. D68, 0300(003), hep-ex/ [3] P. Krokovny, et al., BELLE collaboration, Phys. Rev. Lett. 9, 600(003), hepex/ [4] E. Vaandering, et al., Focus collaboration, hep-ex/ [5] S. Eidelman, et al., (Particle Data Group), Phys. Lett. 59, (004). [6] Y. Mikami, et al., BELLE collaboration, Phys. Rev. Lett. 9, 000(004), hepex/ [7] B. Aubert, et al., BaBar collaboration, Phys. Rev. D69, 030(004), hep-ex/ G. Calderini, et al., BaBar collaboration, hep-ex/ [8] A. Evdokimov, et al., Selex collaboration, hep-ex/ [9] S. Godfrey and N. Isgur, Phys. Rev. D3, 89(985); S. Godfrey and Kokoski, Phys. Rev. D43, 679(99). [0] D. Ebert, V. Galkin and R. Faustov, Phys. Rev. D57, 5663(998), Erratum-ibid, D59, 0990(999), hep-ph/9738. [] R. Lewis and R. Woloshyn, Phys. Rev. D6, 4507(000), hep-lat/ [] J. Hein, et al., Phys. Rev. D6, (000), hep-lat/ [3] N. Isgur and M. Wise, Phys. Lett. B3, 3(989); Phys. lett. B37, 57(990); M. Voloshin and M. Shifman, Yad. Fiz. 45, 463(987); Yad. Fiz. 47, 80(988); B. Grinstein, Nucl. Phys. B339, 53(990); E. Eichten and B. Hill, Phys. Lett. B34, 5(990); A. Falk, H. Georgi, B. Grinstein and M. Wise, Nucl. Phys. B343, (990). [4] M. Nowak, M. Rho and I. Zahed, Phys. Rev. D48, 4370(993), hep-ph/9097; W. Bardeen and C. Hill, Phys. Rev. D49, 409(994), hep-ph/930465; A. Deandrea, N. Bartolomeo, R. Gatto, G. Nardulli and A. Polosa, Phys. Rev. D58, (998), hep-ph/980308; A. Hiorth and J. Eeg, Phys. Rev. D66, 07400(00), hep-ph/ [5] W. Bardeen, E. Eichten and C. Hill, Phys. Rev. D68, 05404(003), hep-ph/ ; M. Nowak, hep-ph/04077; [6] Yu. Kalashnikova and A. Nefediev, Phys. Lett. B49, 9(000), hep-ph/00084; Yu. Kalashnikova, A. Nefediev and Yu. Simonov, Phys. Rev. D64, 04037(00), hepph/00374; Yu. kalashnikova and A. Nefediev, Phys. Lett. B530, 7(00), hep-ph/0330; 8

9 [7] E. Kolomeitsev and M. Lutz, Phys. Lett. B58, 39(004), hep-ph/030733; M. Sadzikowski, Phys. Lett. B579, 39(004), hep-ph/ ; Yuan-Ben Dai, Chao-Shang Huang, Chun Liu and Shi-Lin Zhu, Phys. Rev. D68, 40(003), hep-ph/ [8] T. Barnes, F. Close and H. Lipkin, Phys. Rev. D68, (003), hep-ph/030505; A. Szczepaniak, Phys. Lett. B567, 3(003), hep-ph/ ; G. Bali, Phys. Rev. D68, 0750, hep-ph/ [9] K. Terasaki, Phys. Rev. D68, 050(003), hep-ph/ [0] R. Cahn and J. Jackson, Phys. Rev. D68, 03750(003), hep-ph/0305; P. Colangelo and F. Fazio, Phys. Lett. B570, 80(003), hep-ph/030540; A. Dougall, et al., UKQCD collaboration, Phys. Lett. B569, 4(003), hep-lat/ [] A. Datta and P. O Donnell, Phys. Lett. B567, 73(003), hep-ph/ [] T. Brouder, S. Pakvasa and A. Petrov, Phys. Lett. B578, 365(004), hep-ph/ [3] L. Maiani, F. Piccinini, A. Polosa and V. Riquer, hep-ph/ [4] Yu-Qi Chen and Xue-Qian Li, hep-ph/ [5] Kuang-Ta Chao, hep-ph/040709; T. Barnes, F. Close, J. Dudek, S. Godfrey and E. Swanson, hep-ph/ [6] B. Nicolescu and J. de. Melo, hep-ph/ ; Y.-R. Liu, Shi-Lin Zhu, Y.-B. Dai and C. Liu, hep-ph/ [7] P. Colangelo, F. De. Fazio and R. Ferrandes, hep-ph/040737; M. Nowak, hep-ph/04077; F. De. Fazio, hep-ph/ [8] G. Chew and S. Frautschi, Phys. Rev. Lett. 8, 4(96). [9] S. Filipponi, G. Pancheri and Y. Srivastava, Phys. Rev. Lett, 80,838(998), hepph/97050; S. Filipponi and Y. Srivastava, Phys. Rev. D58, 06003(998), hep-ph/9704. [30] A. Anisovich, V. Anisovich and A. Sarantsev, Phys. Rev. D6, 0550(000), hepph/00033; A. Badalian, B. Bakker and Yu. Simonov, Phys. Rev. D66, 03406(00), hep-ph/ [3] Yu. Simonov, Phys. Lett. B8, 43(989); L. Burakovsky, T. Goldman and L. Horwitz, Phys. Rev. D56, 74(997), hep-ph/ ; R. Fiore, L. Jenkovszky, F. Paccanoni and A. Prokudin, hep-ph/

10 [3] L. Burakovsky, Phys. Rev. D58, (998); A. Kaidalov and Yu. Simonov, Phys. Atom. Nucl. 63, 48(000), hep-ph/999; F. Llanes-Estrada, S. Cotanch, P. Bicudo, J. Ribeiro and A. Szczepaniak, Nucl. Phys. A70, 45(00), hep-ph/0008; M. Brisudova, L. Burakovsky, T. Goldman and A. Szczepaniak, Phys. Rev. D67, 09406(003), nucl-th/03030; H. Meyer and M. Teper, Nucl. Phys. B668, (003), hep-lat/030609; L. Zayas, J. Sonnenschein and D. Vaman, Nucl. Phys. B68, 3(004), hep-th/0390. [33] Yu. Yufryakov, hep-ph/ [34] A. Tang and W. Norbury, Phys. Rev. D6, 06006(000), hep-ph/ ; M. Brisudova, L. Burakovsky and T. Goldman, Phys. Rev. D6, 05403(000), hepph/ [35] A. Inopin and G. Sharov, Phys. Rev. D63, 05403(00), hep-ph/ ; A. Badalian and B. Bakker, Phys. Rev. D66, 03405(000), hep-ph/0046; Yu. Simonov, Phys. Atom. Nucl. 66, 038(003), hep-ph/ [36] N. Isgur, Phys. Rev. D57, 404(998). [37] K. Abe, et al., Belle collaboration, Phys. Rev. D69, 00(004), hep-ex/ [38] P. Abreu, et al., DELPHI collaboration, Phys. Lett. B46, 3(998). [39] D. Melikhov and O. Pene, Phys. Lett. B446, 336(999), hep-ph/ ; P. Page, Phys. Rev. D60, 05750(999), hep-ph/