Blejske delavnice iz fizike Letnik 12, št. 1 Bled Workshops in Physics Vol. 12, No. 1 ISSN 1580-4992 Proceedings of the Mini-Workshop Understanding hadronic spectra Bled, Slovenia, July 3 -10, 2011 Edited by Bojan Golli Mitja Rosina Simon Sirca University of Ljubljana and Jozef Stefan Institute DMFA - ZALOŽNIŠTVO Ljubljana, november 2011 The Mini-Workshop Understanding hadronic spectra was organized by Jožef Stefan Institute, Ljubljana Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana and sponsored by Slovenian Research Agency Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Organizing Committee Mitja Rosina, Bojan Golli, Simon ¡Sirca List of participants Luis Alvarez Ruso, Valencia, alvarez@teor. fis .uc.pt Enrique Ruiz Arriola, Granada, earriola@ugr. es Wojtek Broniowski, Krakow, b4bronio@cyf-kr.edu.pl Marko Bračko, Ljubljana, marko. bracko@ij s.si Joseph Day, Graz, j day21@gmail. com Veljko Dmitrašinovic, Belgrade, dmitrasin@yahoo. com Bojan Golli, Ljubljana, boj an. goll i@ij s.si Regina Kleinhappel, Graz, regina . kleinhappel@gmx. at Daniel Kupelwieser, Graz, daniel. kupelwieser@uni-graz . at Willi Plessas, Graz, willibald.plessas@uni-graz.at Martin Rohrmoser, Graz, martin.rohrmoser@edu.uni-graz .at Saša Prelovšek, Ljubljana, Sasa . Prelovsek@ij s.si Mitja Rosina, Ljubljana, mitj a . rosina@ij s.si Wolfgang Schweiger, Graz, wolfgang. schweiger@uni-graz . at Vikram Soni, New Delhi, v. soni@airtelmail. in Ica Stancu, Liege, fstancu@ulg .ac.be Simon Sirca, Ljubljana, simon.sirca@fmf.uni-lj .si Electronic edition http://www-f1.ijs.si/BledPub/ Contents Preface............................................................. V Predgovor..........................................................VII Meson dynamics in the vector-scalar sector L. Alvarez-Ruso, J. A. Oller, J. M. Alarcon............................... 1 0++ states in a large-Nc Regge approach Enrique Ruiz Arriola and Wojciech Broniowski.......................... 7 Transversity structure of the pion in chiral quark models W. Broniowski, E. Ruiz Arriola, A. E. Dorokhov.......................... 18 Spectroscopy of heavy baryons Joseph P. Day, Ki-Seok Choi, Willibald Plessas........................... 25 Baryons' anomalous magnetic moments in a UL(3) x UR (3) chiral symmetric theory V Dmitrasinovic, Hua-Xing Chen, Atsushi Hosaka....................... 32 On the way to a realistic description of hadron resonances Regina Kleinhappel, Willibald Plessas, Wolfgang Schweiger............... 36 The Schwinger model in point form D. Kupelwieser, W. Schweiger, and W. H. Klink.......................... 39 Electroweak structures of light and strange baryons Ki-Seok Choi and W. Plessas.......................................... 44 Analysis of flavor contributions to electromagnetic nucleon form factors M. Rohrmoser, Ki-Seok Choi, W. Plessas................................ 47 A review of new data on compact stars from a Magnetar model of magnetised cores Vikram Soni and N. D. Haridass....................................... 55 Negative parity nonstrange baryons in large Nc QCD: quark excitation versus meson-nucleon scattering N. Matagne and Fl. Stancu............................................ 57 News from Belle M. Bracko........................................................... 64 Electroproduction of mesons in a chiral quark model B. Colli............................................................. 70 Scattering phase shift and resonance properties S. Prelovsek, C. B. Lang and D. Mohler................................. 73 The pion cloud of the nucleon in the constituent quark picture Bogdan Povh and Mitja Rosina........................................ 82 Recent experimental results from MAMI (Mainz), ELSA (Bonn), and JLab S. Sirca............................................................. 88 Preface With the MiniWorkshop behind us, do we now better understand hadronic spectra? Or rather, will these Proceedings remind you that at our meeting, as many new problems have been opened as old ones resolved? But it is clear that we made progress on several fronts, and have learned a lot from each other. Electroexcitation of baryons continues to be a fruitful source of information about the structure and spectra of light baryons. It is encouraging that many ideas, especially about pion clouds, are applicable also at higher energies, in deep inelastic scattering. The role of relativity has been further documented in understanding strange baryon spectra. The problem of theoretically underestimated resonance widths persists, while coupled-channels approaches instill some hope in its ultimate resolution. Lattice QCD has become a respectable operational theory, and now even quark modelists can resort to it to obtain some insights, for example, into pion-pion scattering or transverse structure of the pion. The 1/Nc expansion has also gained some appreciation by suggesting, among other things, a SU(2Nf) classification of baryon multiplets (for Nc colours and Nf flavours). This expansion also helps to understand scalar mesons and their mixing with glueballs. New resonances in the charmonium spectrum still excite interaction between experimentalists and theorists (as much as the interaction between dimeson and tetraquark configurations). Our eyes should thus also be aimed at Heaven: perhaps new surprises about stability of 3-quark clusters in dense nuclear matter are hidden in the clues provided by the heavy magnetars? Therefore, we need you at Bled again, next year. Ljubljana, November 2011 M. Rosina B. Golli S. Sirca Predgovor Ko se oziramo nazaj na letošnjo Blejsko delavnico iz fizike, ali se nam zdi, da sedaj bolje razumemo hadronske spektre? Ali pa nas bo ta Zbornik opomnil, da smo na nasem srečanju odprli vsaj toliko novih problemov, kot smo resili starih? Jasno pa nam je, da smo napredovali na mnogih frontah in da smo se veliko naučili drug od drugega. Ta predgovor v slovensčini naj sluzi tudi kot kratek povzetek nasih aktivnosti in dosezkov. Vzbujanje barionov s trki elektronov je se vedno ploden vir informacij o zgradbi in spektrih lahkih barionov. Vzpodbudno je, da veljajo mnoge ideje, zlasti o pion-skih oblakih, tudi pri visokih energijah pri globoko neelastičnem sipanju. Vedno močneje se zavedamo vloge relativnosti, ki se je izkazala kot neizogibni sestavni del pristopov k razumevanju spektrov čudnih barionov. Vseeno nas se vedno skrbijo problemi s sirinami resonanč, saj enostavni teoretični modeli napovedujejo premajhne sirine; resitev pričakujemo sele od računov s sklopljenimi razpadnimi kanali. Kromodinamika na mrezi si je pridobila ugled, da vsaj kvalitativno uspesno simulira lastnosti marsikaterih količin. Celo mojstri kvarkovskih modelov lahko iz nje dobijo koristne vpoglede, rečimo v sipanje piona na pionu ali v transverzalno zgradbo piona. Razvoj po rečipročnem stevilu barv je bil tudi odmeven. Med drugim je sugeriral klasifikačijo barionskih multipletov z grupo SU(2Nf) v zvezi z okusi kvarkov, pomagal pa je tudi razumeti mesanje skalarnih mezonov med seboj in z gluonijem. Nove resonanče v spektru čarmonija se vzpodbujajo sodelovanje med eksperi-mentalči in teoretiki (kakor tudi med dimezonskimi in tetrakvarkovskimi konfig-uračijami). Ozirati pa se moramo tudi v nebo: morda nas čakajo presenečenja o stabilnosti trokvarkovskih gruč znotraj goste jedrske snovi, ki jih skrivajo tezke magnetne zvezde, magnetarji. Torej vas drugo leto spet potrebujemo na Bledu! Ljubljana, novembra 2011 M. Rosina B. Golli S. Sirca Workshops organized at Bled > What Comes beyond the Standard Model (June 29-July 9,1998), Vol. 0 (1999) No. 1 > Hadrons as Solitons (July 6-17,1999) > What Comes beyond the Standard Model (July 22-31,1999) > Few-Quark Problems (July 8-15, 2000), Vol. 1 (2000) No. 1 > What Comes beyond the Standard Model (July 17-31, 2000) > Statistical Mechanics of Complex Systems (August 27-September 2, 2000) > Selected Few-Body Problems in Hadronic and Atomic Physics (July 7-14,2001), Vol. 2 (2001) No. 1 > What Comes beyond the Standard Model (July 17-27,2001), Vol. 2 (2001) No. 2 > Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry > Quarks and Hadrons (July 6-13, 2002), Vol. 3 (2002) No. 3 > What Comes beyond the Standard Model (July 15-25,2002), Vol. 3 (2002) No. 4 > Effective Quark-Quark Interaction (July 7-14, 2003), Vol. 4 (2003) No. 1 > What Comes beyond the Standard Model (July 17-27, 2003), Vol. 4 (2003) Nos. 2-3 > Quark Dynamics (July 12-19, 2004), Vol. 5 (2004) No. 1 > What Comes beyond the Standard Model (July 19-29,2004), Vol. 5 (2004) No. 2 > Exciting Hadrons (July 11-18, 2005), Vol. 6 (2005) No. 1 > What Comes beyond the Standard Model (July 18-28,2005), Vol. 6 (2005) No. 2 > Progress in Quark Models (July 10-17, 2006), Vol. 7 (2006) No. 1 > What Comes beyond the Standard Model (September 16-29, 2006), Vol. 7 (2006) No. 2 > Hadron Structure and Lattice QCD (July 9-16, 2007), Vol. 8 (2007) No. 1 > What Comes beyond the Standard Model (July 18-28,2007), Vol. 8 (2007) No. 2 > Few-Quark States and the Continuum (September 15-22,2008), Vol. 9 (2008) No. 1 > What Comes beyond the Standard Model (July 15-25, 2008), Vol. 9 (2008) No. 2 > Problems in Multi-Quark States (June 29-July 6, 2009), Vol. 10 (2009) No. 1 > What Comes beyond the Standard Model (July 14-24, 2009), Vol. 10 (2009) No. 2 > Dressing Hadrons (July 4-11, 2010), Vol. 11 (2010) No. 1 > What Comes beyond the Standard Model (July 12-22, 2010), Vol. 11 (2010) No. 2 > Understanding hadronic spectra (July 3-10,2011), Vol. 12 (2011) No. 1 > What Comes beyond the Standard Model (July 11-21, 2011), Vol. 12 (2011) No. 2 Also published in this series > Book of Abstracts, XVIIIEuropean Conference on Few-BodyProblems in Physics, Bled, Slovenia, September 8-14,2002, Edited by Rajmund Krivec, Bojan Golli, Mitja Rosina, and Simon Sirca, Vol. 3 (2002) No. 1-2 Bled Workshops in Physics Vol. 12, No. 1 p.l Meson dynamics in the vector-scalar sector* L. Alvarez-Rusoa, J. A. Ollerb, J. M. Alarconb a Instituto de Física Corpuscular, UVEG-CSIC, Valencia, Spain b Departamento de Física, Universidad de Murcia, Murcia, Spain Abstract. We have studied the ^(1020)f0(980) and $(1020)a0 (980) S-wave scattering at threshold energies employing chiral Lagrangians coupled to vector mesons by minimal coupling. The ^ f0 a0) interaction kernel is obtained by treating the f0 (980) [a0(980)] as bound (dynamically generated) state and resuming unitarity loops. We are able to describe the e+ e- —> ^(1 020)f0 (980) recent scattering data concluding that the Y(2175) resonance has a large ^(1020)f0(980) component. We also predict a strong ^(1020)a0(980) interaction that can be studied in e+ e- —> ^rt0r|. For some sets of parameters a clear resonant peak indicates the presence of an isovector companion of the Y(2175). 1 Introduction A new hadronic spectroscopy has emerged in the last decade thanks to the experimental activity carried out meanly at e+e- facilities (BES at IHEP, CLEO at LEPP, BABAR in SLAC, Belle at KEK) but also at pp colliders (CDF,D0 at FNAL) and in fixed target experiments such as HERA-B at DESY. Indeed, our understanding of meson spectroscopy has been challenged by the observation of several exotic states (extensive reviews can be found, for example, in Ref. [1]). These can be neutral mesons with quantum numbers that are not allowed for qq pairs (JPC = 0 , 0+-, 1 +, 2+ ,...) but also states with conventional quantum numbers that cannot be easily accommodated into the constituent quark model. One such a state is the resonance ^(2170) (or Y(2175), as we will refer to it from now on), a light unflavored meson with quantum numbers JPC = 1 , IG = 0-, mass of 2175 ± 15 MeV and width rY = 61 ± 18 MeV (PDG estimates [2]). It was first observed by the BABAR Collaboration [3,4] in the initial-state radiation process e+e- —> ^f0(980) y —> K+K- nny and also found by BES in J/¥ —> n ^ f0 (980) decay [5]. The Belle Collaboration has performed the most precise measurements so far of the reactions e+e- —> ^ n+n- and e+e- —> ^ f0(980) finding My = 2079 ± 13-79 MeV and rY = 192 ± 23+65 MeV [6]. The obtained width is larger than in previous measurements but the error is also large. These experimental findings have triggered a considerable theoretical activity aimed at unraveling the nature and properties of the Y(2175). It has been interpreted as a tetraquark [7-9], with a mass of 2.21 ± 0.09 GeV [7] or 2.3 ± 0.4 GeV [8] calculated using QCD sum rules with meson-meson (ss)(ss) currents [7] and * Talk delivered by L. Alvarez-Ruso adding diquark-antidiquark (ss)(ss) ones [8]. In the diquark-antidiquark picture a prominent Y(2175) —> AA decay mode appears [9]. The Y(2175) has also been identified with the lightest hybrid ssg state [10] with Ki (1400) K and Ki (1270) K as dominant decay channels. Conventional ss states in 23D1 or 33S1 configurations have been considered as their masses are expected to be compatible with the Y(2175) [11] although the estimated widths are too large. Reference [12] studies the three-body KK^(1020) scattering with two-body pseudoscalar-pseudoscalar and vector-pseudoscalar interactions taken from unitarized chiral perturbation theory [13,14]. A resonance with 2170 MeV mass is generated albeit with a width of only 20 MeV. 2 0(1020) fo(980) scattering In Ref. [15] we have studied the S-wave scattering of the vector meson ^(1020) with the scalar f0 (980), the channel with the same quantum numbers as the Y(2175). This is feasible because both the ^(1020) and the f0(980) are rather narrow resonances. First we derive the kernel of the ^ f0 interaction. For this we take advantage of the fact that the f0(980) scalar meson is successfully described as a KK bound state [13,16]. This means that in the second Riemann sheet, in the vicinity of the fo(980) pole ~iTKK = v2 TL +yi+T2(M^-k2)+... , and lim, (M2o-k2)(-iTKK) =y20. k2 — M2 0 k2-> m2 0 to f 0 (1) Therefore, the ^(1020)f0(980) interaction can be obtained from the ^(1020)KK one by extracting the residue at the f0(980) double pole position that arises from the initial and final KK rescatterings. Fig. 1. Feynman diagrams for ^KK scattering. Dashed lines denote kaons and solid ones, vector mesons. 2 3 4 6 7 10 8 9 12 14 15 17 13 16 The contributions to the ^(KK)I=0 —> ^(KK)I=0 amplitude, determined with chiral Lagrangians coupled to vector mesons are depicted in Fig. 1. It can be shown [15] that close to the ^KK threshold and taking into account that the fo(980) is also close to the KK threshold, the dominant term is given by diagram 2. The rescattering of initial and final KK pairs in this dominant amplitude gives rise to the diagram on the left hand side of Fig. 2. For the (KK)2 vertices we take f0(980) f0(98g) -'J.''-- VX fo (980) Vf 0 ( 98 0) | _ Fig.2. Dominant contribution to the ^(KK)i=o amplitude with KK initial and final state interactions that contain f0 (980) poles. + only on-shell amplitudes. The off-shell parts are proportional to the inverse of kaon propagators and cancel with them in the calculation of the loop, resulting in amplitudes that do not correspond anymore to the dominant triangular kaon-loop but to other topologies. After projecting into S-waves MS=o =-t*K Tkk(k2) Tkk(k'2) Ls (2) where t^K and Tkk are the full scattering amplitudes, k2(k'2) is the initial (final) KK invariant mass and Ls with 1 4T? '+1 d cos p i •1/2 ! dx- [log (1 - Ix/c) - log (1 + 2x/c)] , (3) o c 2 _ 4 l"„2o2 C' = — [x2Q2 + 2k2x(l -2x) +ie] . (4) Q2 Here Q2 = —2p2(1 — cos p) in terms of the relative angle p between the incoming p and outgoing p' ^ three-momenta in the ^f0 CM frame. The residue at the fo (980) double pole is the fo(980)^(1020) potential To kMc'^M* ^ lim Jk2-M2o)(k'2-M2o)Mf=0=-t^Ky^LS) (5) which is unitarized as schematically shown in Fig. 3 leading to the full ^f0 amplitude n*fo=T—T7-p-• (6) I + Vf Gf The loop function Gf is expressed in terms of a renormalization scale fixed to the p meson mass = 770 MeV and a subtraction constant a to be fitted to data [15]. We have performed fits to the e+e- ^f0(980) BABAR and Belle data [4, 6]. The ^(1020) f0 (980) strong scattering amplitude is employed to correct the production process by final state interactions (FSI) ! \ CTBG (s) ct(sJ =-j . (7) 11 + Vf G^fo1 + (1) (2) Fig. 3. Diagrammatic representation of the full ^f0 (980) amplitude. For the nonresonant background production cross section the Belle fit (Fig. 6(b) of Ref. [6]) has been adopted. In our fits the fo(980) properties, pole position Mfo and residue y2 are taken from two different studies [17,18]; and ai are free parameters. The results are presented in Table 1 and Fig. 4 Mf0 [MeV] (fixed) y2 [GeV2 ] (fixed) td,f0 ai Fit 1 980 16 -54 ± 4 -2.41 ± 0.14 Fit 2 988 13.2 -27 ±1 -2.61 ±0.14 Table 1. Fits to the e+ e" 4>(1020)fo(980) BABAR [4] and Belle [6] data. to fo to to to to to 0 V^[GeV] Fig. 4. Cross section for e+ e —> <^(1020) fo (980). The experimental data are from Ref. [4] (diamonds and crosses) and Ref. [6] (empty boxes). The solid and dash-dotted lines correspond to the first and second fits of Table 1. The dashed line shows the background. The description of the data is satisfactory, particularly the peak position and width. Worse is the agreement at a/s < 2 GeV: the suppression of the theoretical curves happens because the Vf potential is large due to the 1/Q2 factor. We obtain negative values for ai as it should be for a dynamically generated resonance. Moreover, the resulting scale A = [Ani)/~ 0.75 GeV, preserves a natural size around Mp. The interpretation of the t^K values is more difficult due to the lack of information about the ^K interaction close to threshold. Nevertheless one should recall that the Ki (1400) resonance is only 100 MeV below this threshold. Therefore, the assumption that ^K scattering is dominated by the K1 ( 1400) would explain the negative sign of t^K because Our fitted t^K values are very different from those used in Ref. [12], taken from Ref. [14] which does not contain the K1 ( 1400). With a t^K - 12 - 7 i as in Ref. [14] we would not describe the e+ e- —> ^(1020) f° (980) data. This means that even if the results of both Refs. [12,15] support the interpretation of the Y(2175) as a dynamically generated resonance, the two descriptions are quantitatively different. 3 0(1020) ao(980) scattering In the present contest, it is relevant to establish whether there is an isovector companion of the isoscalar Y(2175). It will help constraining theoretical models. In particular, the calculation of Ref. [12] does not find any resonance in the isovector ^ (1020) a0 (980) S-wave channel. Experimentally, this resonance could show up in e+ e- ^(1020)a0(980) ^ ^(1020)non, as suggested in Ref. [19] or in e+ e-^(1020)a0(980) —> ^(1020)K+ K- [20]. Our study of the ^(1020)a0(980) [21] interaction proceeds as described in the previous section but replacing the scalar-isoscalar f0(980) by the scalar-isovector a0(980). The latter is treated as a dynamical resonance in coupled channels (mainly KK and n0n) whose properties depend on the adopted approach (see Table 2). No new free parameters need to be introduced if one demands that the e+ e- —> ^(1020) f0 (980) cross section is reproduced and takes t^K, ai from Table 1. _Mao [GeV] Ykr [GeV2] BS 1.009 + i 0.056 24.73 - i 10.82 N/D 1.055 +i 0.025 17.37-i 24.77 Table 2. a0(980) properties, pole position Mao and residue yKk, as obtained with the Bethe-Salpeter (BS) equation [13] and the N/D method [18]. We have investigated the corrections to the e+e- —> ^(1020) a° (980) —> ^(1020)n°n reaction that arise from ^(1020) a°(980) FSI finding strong modifications (see Fig. 5). If the a°(980) properties from the N/D method are taken, a strong peak around 2.03 GeV is observed, signaling the presence of the dynamically generated isovector 1 resonance. For the BS pole no peak is generated but a strong reduction of the cross-section takes place. This result further supports the idea that a study of the e+ e- —> ^ ( 1020) a° (980) reaction, which should be accessible at present e+ e- factories, may provide novel relevant information about hadronic structure and interactions in the 2 GeV region. Fit 1 Fit 2 v^ [GeV] [GeV] Fig.5. e+ e- -> $(1020)a°(980) -> ^(1020)n°r| cross section. The dotted lines in both plots is the result of Ref. [19] where final state ^(1020)a° (980) rescattering was not considered. The rest of the lines include FSI for the sets of parameters given in Tables 1,2. References 1. S. L. Zhu, Int. J. Mod. Phys. E 17 (2008) 283; N. Drenska et al., Riv. Nuovo Cim. 033 (2010) 633-712; N. Brambilla et al, Eur. Phys. J. C71 (2011) 1534. 2. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667 (2008) 1. 3. B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 74 (2006) 091103. 4. B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 76 (2007) 012008. 5. M. Ablikim, et al. [BES Collaboration], Phys. Rev. Lett. 100 (2008) 102003. 6. C. P. Shen et al. [Belle Collaboration], Phys. Rev. D 80 (2009) 031101. 7. Z. G. Wang, Nucl. Phys. A 791 (2007) 106. 8. H.-X. Chen, X. Liu, A. Hosaka and S.-L. Zhu, Phys. Rev. D 78 (2008) 034012. 9. N. V. Drenska, R. Faccini and A. D. Polosa, Phys. Lett. B 669 (2008) 160. 10. G.-J. Ding and M.-L. Yan, Phys. Lett. B 650 (2007) 390. 11. G.-J. Ding and M.-L. Yan, Phys. Lett. B 657 (2007) 49; T. Barnes, N. Black and P. R. Page, Phys. Rev. D 68 (2003) 054014. 12. A. Martinez Torres et al, Phys. Rev. D 78 (2008) 074031. 13. J. A. Oller and E. Oset, Nucl. Phys. A 620 (1997) 438; (E)-ibid. A 652 (1999) 407. 14. L. Roca, E. Oset and J. Singh, Phys. Rev. D 72 (2005) 014002. 15. L. Alvarez-Ruso, J. A. Oller and J. M. Alarcon, Phys. Rev. D 80 (2009) 054011. 16. J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48 (1982) 659. 17. M. Albaladejo and J. A. Oller, Phys. Rev. Lett. 101 (2008) 252002. 18. J. A. Oller and E. Oset, Phys. Rev. D 60 (1999) 074023. 19. C. A. Vaquera-Araujo and M. Napsuciale, Phys. Lett. B 681 (2009) 434. 20. S. Gomez-Avila, M. Napsuciale and E. Oset, Phys. Rev. D 79 (2009) 034018. 21. L. Alvarez-Ruso, J. A. Oller and J. M. Alarcon, Phys. Rev. D 82 (2010) 094028. Bled Workshops in Physics Vol. 12, No. 1 P. 7 0++ states in a large-N c Regge approach* Enrique Ruiz Arriolaa and Wojciech Broniowski b a Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain b The H. Niewodniczanski Institute of Nuclear Physics PAN, PL-31342 Krakow, Poland, and Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland Abstract. Scalar-isoscalar states (JPC = 0++) are discussed within the large-N c Regge approach. We find that the lightest f° (600) scalar-isoscalar state fits very well into the pattern of the radial Regge trajectory where the resonance nature of the states is advantageously used. We confirm the obtained mass values from an analysis of the pion and nucleon spin- 0 gravitational form factors, recently measured on the lattice. We provide arguments suggesting an alternating meson-glueball pattern of the 0++ states, which is supported by the pseudoscalar-isovector 0+- excited spectrum and asymptotic chiral symmetry. Finally, matching to the OPE requires a fine-tuned mass condition of the vanishing dimension-2 condensate in the Regge approach with infinitely many scalar-isoscalar states. 1 Introduction The goal of this talk is to discuss various intriguing aspects of the spectrum of scalar-isoscalar states. Approaches developed in recent years may shed new light on this long-elaborated problem in hadronic physics. Subsequent hadron resonances listed in the Particle Data Group (PDG) tables increase their mass up to the upper experimental limit of 2.5 GeV, while their width remains bound within 500 MeV. In Fig. 1 we show separately the widths of all baryons and mesons listed in the PDG tables [1] as functions of the mass of the state. One naturally expects that broad resonances, i.e., with r ~ m, escape phenomenological analysis; even if they existed, they might be missing from the PDG as difficult to assess experimentally. Note, however, that with the exception of the notorious f0(600) resonance and a few baryon and meson states, the ratio is bound by the line r/m ~ 1 /3 (the dashed line in Fig. 2). A natural and model-independent framework to understand this feature is provided by the limit of large number of colors in QCD. Indeed, in this large-Nc limit, with g2Nc fixed, baryons are heavy with mass m = O(Nc) and width r = O(N°) [2,3], while mesons and glueballs are stable, with mass independent of Nc, namely m = O(N°), and width r suppressed as 1 /Nc and 1 /N;:, respectively. This means that r/m is suppressed (see, e.g., [4] for a review). In particular, one has r/m ~ N-1 for mesons and baryons, while r/m ~ N-2 for glueballs.1 * Talk delivered by Enrique Ruiz Arriola 1 Fig. 1 suggests that it is reasonable to assume that excited states in the spectrum follow a more accurate large-N c pattern than the ground state. > M2 (GeV2) Fig. 3. Radial Regge trajectory corresponding to the squared mass of all JPC = 0++ scalar-isoscalar and JPC = 0-+ pseudoscalar-isovector states listed in the PDG tables [1]. The four heaviest 0++ and two 0-+ states are not yet well established and are omitted from the PDG summary tables. The error bars correspond to the errors in the determination of the square of mass as Am2 = mF with F from [1]. The straight line is the result of our joined fit. Labels of 0++ states are above their mark whereas labels of 0+- states are below their mark. Minimization yields x2/DOF = 0.12, with a = 1.31(12) GeV2, mff = 556(127) MeV. (4) Formula (2) is actually equivalent to two parallel radial Regge trajectories with the standard slope, Ms,_(n)2 = an + m; Ms,+ (n)2 = an + m£ + 2 (5) (6) where a = 2na, and ct is the string tension associated to the potential V(r) = or between heavy colored sources. The value a/o1 = 456(21) MeV obtained from our fit agrees well with lattice determinations of = 420 MeV [19]. Of course, one expects some of these states to correspond eventually to glueballs. However, there seems to be no obvious difference between mesons and glueballs, as far as the radial Regge spectrum is concerned. Note that Casimir scaling suggests that the string tension is o"giuebaii = f Cmeson/ but this holds in the case of fixed and heavy sources. The fact that we have light quarks might explain why we cannot allocate easily the Casimir scaling pattern in the light-quark scalar-isoscalar spectrum. a 4 Interpolating fields For scalar states a measure of the spectrum is given in terms of the (gauge and renorm invariant) trace of the energy momentum tensor [20] ©[; = © = J^Ig^G^ + Y_ ™q [1 + Ym(a)] qq. (7) q Here (3(a) = |x2da/d^.2 denotes the beta function, a = g2/(4n) is the running coupling constant, Ym(a) = dlogm/dlog |j.2 is the anomalous dimension of the current quark mass mq, and is the field strength tensor of the gluon field. It is interesting to consider the situation of massless quarks, where things become somewhat simpler. Then, we have in principle two scalar operators with smallest canonical dimensions, the gluon G2 and the quark qq. While these two operators are both scalars, they are chirally even and odd, respectively, i.e., under the q —> Y5 q transformation. Because the chiral symmetry is spontaneously broken, there is some mixing between G2 and qq. These operators connect scalar states to the vacuum through the matrix element (0|©|n) = mf , (0|qq|n) = m^c^ . (8) The two-point correlators read n00(q) = i n@s(q) = i nss(q) = i f2 m4 d4xeiqx(O|T{0(x)0(O)}|O) = Y 7n n 7+c.t., (9) V mn - q2 f2 m2 c m d4xeiq x(O|T{0(x)qq(O)}|0) = +c t > (10) n n 22 d4xeiq "(0|T{qq(x)qq(0)}|0) = £ "T^ +c-t- - (H) mn q where in the r.h.s. we saturate with scalar states and c.t. stands for subtraction constants which can chosen as to replace m2 —> q2 in the numerator. In that scheme, in the large — q2 > Aqcd limit, a comparison with the Operator Product Expansion (OPE) [21-23] leads to the matching conditions n00(q2) = q4 Co log(—q2)+ ..., nss(q2) = q2 CO log(—q2) + ..., n0s(q2) = CO' (q q) log(—q2) + ..., (12) where C0 = — (2p(a)/an)2, CO = —3/(8n2) and CO' = —2p(a)/an. As we see, qq and G2 do not mix at high q2 values, a consequence of asymptotic chiral symmetry [20]. In these limits the sums over n can be replaced by integrals, whence the following asymptotic conditions are found: f2/(dm2/dn) Co , c2/(dm2/dn) CO , Cnfnm2/(dm2/dn) C'(qq(13) We see that the first two conditions are incompatible with the third one if m2 increases for large n, as is the case of the data. However, if we group the states in two families, as suggested by Eqs. (6) and Fig. 3, we get a compatible solution c2)-,f2)+—> const c2)+,f2>_—> const/m^. (14) This is equivalent to assuming an asymptotically alternating pattern of mesons and glueballs, coupling to chirally odd and even operators, q q and G2, respec- 1 3 tively. Since asymptotically m^ - an/2, we find cn _/cni+ and fn,+/fn,- ^ n.?. Of course, this is not the only solution. The situation described above suggests the existence of a hidden symmetry in the 0++ sector. In our case we could think of the y5-parity (which becomes a good quantum number for excited states) as the relevant symmetry which makes the difference between the chirally even and odd states. This, however, only explains the fact that asymptotically the slopes of the + and — branches are the same, but not why the intercepts accurately differ by half the slope. 5 The holographic connection To further elaborate on this intriguing point of the accidental degeneracy, let us consider the one-dimensional harmonic oscillator with frequency as an example; all states ^n(z) with the energy En = h^(n + 1 /2) can be separated into parity even and parity odd states, satisfying the conditions ^n,± (z) = ^2n(z) and ^±,n( — z) = ±^±,n(z), respectively, and having the energies E+,n = /4) and E_,n = + 3/4). These formulas display twice the slope of En. Thus, given the states with energies E+,n and E_,n, we might infer that parity was a hidden symmetry of a Hamiltonian explaining the correlation between the slope and intercepts. In the relativistic case the argument can also be made in a suggestive manner. Let us consider the Klein-Gordon action for infinitely many bosons in four dimensions, described with fields ^n(x) of masses mn: d4x ^ - mM • (15) We assume the spectrum of the form m^ = an + m°. Next, we can introduce the five-dimensional field ^(x,z) = Y.n ^n(x)^n(z), with ^n(z) fulfilling the auxiliary Sturm-Liouville problem in the variable 0 < z < oo, —9z [p(z)9z^n(z)l + q(z)^n(z) = m^M^M, (16) where the functions are orthogonal with respect to the weight function p(z), provided suitable boundary conditions p(z) (z)^m(z) — ^niz^m(z)) |z=0 = 0 (17) and ^n(oo) —> 0 are fulfilled. The action can then be written as 1 S 2 d4 x dz [p(z)9^9^ - p(z)(9^)2 - q(z)^2] (18) after some integration by parts in the variable z. This action can be written as a five-dimensional action with a non-trivial metric [24], featuring the AdS/CFT (soft-wall) approach (see [25] and references therein), with the extra dimension z playing the role of a holographic variable and the orthogonal set of functions ^n(z) denoting the corresponding Kaluza-Klein modes. Clearly, z has the dimension of length, suggesting that z 0 corresponds to the ultraviolet and z to the infrared regime. 0 Turning to Eq. (16), we may take the standard Harmonic oscillator Schrodinger-like equation (p(z) = p(z) = 1, q(z) = U(z) = a2z2/16) + la2z2^n(z) = m2^n(z) (19) and obtain for the regular solutions at infinity the result K(O) WO) v r (!-■=£) (20) where r (x) is the Euler Gamma function, which is meromorphic and have simple poles at x = 0, — 1, —2,____The solutions fulfilling the Dirichlet, ^n(0) = 0, and Neumann, ^n (0) = 0, boundary conditions, respectively, have the masses 2 a 2 3a m-,n = arL+4> m+,n = arL + -j- . (21) which can be merged into one single formula a m2=-(2u + 1). (22) This yields mCT = m,f0/\/3 = 566 MeV and, for the string tension, ct = m,f0 \flfZri = 450 MeV with mf0 = 980 MeV, quite reasonable values. In this approach the symmetry in the scalar spectrum corresponds to a parity symmetry in the holographic z variable z) = ±^n(z). Note that usually the holographic variable z is taken to be positive3, but if we extend it to —«> < z < oo, we may define a holographic superfield containing two different and orthogonal modes. Otherwise, in the interval 0 < z < the Dirichlet and Neumann modes are not orthogonal to each other. 6 Pseudoscalar mesons and chiral symmetry Discerning the nature of the ct state has been a recurrent pastime for many years. As is well known, glueballs are more weakly coupled to mesons, O(1/NC), than other mesons, 0( 1 /a/Nc)- The minimum number of states, allowed by certain sum rules and low energy theorems, is just two. In Ref. [16] we undertake such an analysis which suggests that f0(600) (denoted as ct) is a qq meson, while f0(980) (denoted as f0) is a glueball. This is supported by the rather small width ratio, 3 This is supported by the light-front interpretation of Brodsky and de Teramond [26], where the holographic variable is the polar coordinate of a two dimensional vector, z = |C| and £ = b^xO" — x), with b denoting the impact parameter and x the momentum fraction of the quark. This interpretation yields a two dimensional potential U(Z) = k2 Z2 + 2k2 (L + S — 1) with J = L + S which, when passing to the polar variable z, generates the usual centrifugal term (L2 —1 /4)/z2 not present in our discussion, yielding Min.l.s = 4k2 (n + L + S/2) which for J = 0 and L = 1 resembles Eq. (22). which yields i>/rff - {g^nm3f)/{gl„„m3J - 1/NC, thus for mff - 0.8 MeV - mf the ratio ga7m/9inn - a/Nc is obtained. A further piece of evidence for the alternating meson-glueball pattern is provided by looking at the excited pion spectrum, which we show in Fig. 3. The alternating pattern was unveiled by Glozman [27], suggesting that states degenerate with the pion might not be identified with glueballs. Remarkably, the states generating doublets with pion states are f0(600) n0(140), f0(1370) n0(1300), f0(1710) ^ n0(1800), f0(2100) ^ n0(2070), and f0(2330) ^ n0(2360), whereas the other scalar states f0(980), f0(1500), f0(2020) and f0(2200) are not degenerate with other mesons with light u and d quarks. Our analysis is reinforced by this observation. As a matter of fact, fitting the pion n(140) as the n = 0 state of the Regge spectrum requires strong departure from a simple linear trajectory, m^ = an + m0. One may improve on this by using the holographic connection and a mixed boundary condition at z = 0 determined by fixing the mass of the ground state m0 using Eq. (17) together with Eq. (20) for the harmonic oscillator case, Eq. (19). This procedure ensures the orthogonality between all states and implements linearity for large n. This can be done for the ground states m0 = mn, mCT, mfo. The fit to all states yields a = 1.37GeV2 and the mass spectra (in GeV) n (Regge) (0.140,1.260,1.730, 2.092, 2.400,...) n (PDG) (0.140,1.300,1.812, 2.070, 2.360) a- (Regge) (0.527,1.297,1.750, 2.106, 2.411,...) a (PDG) (0.600,1.350,1.724, 2.103, 2.321) f0 (Regge) (0.977,1.513,1.906, 2.232, 2.517,...) f(PDG) (0.980,1.505,1.992, 2.189) yielding 1 /(0)/t|>0 (0) = —3.1, —14.9, and 0.2, respectively. Note the large and small values for the a and f0 cases, which suggests that these boundary conditions are very close to the Dirichlet and Neumann cases. Chiral symmetry breaking corresponds to the different n and a values. 7 Scalar dominance and heavy pions Hadronic matrix elements of the energy-momentum tensor, the so-called gravitational form factors (GFF) of the pion and nucleon, correspond to a dominance of scalar states in the large-Nc picture, as (u(p) is a Dirac spinor) where the sum rules Y.n gnnnfn = 1 [28] and MN = Y.n gnNNfn [29] hold. Unfortunately, the lattice QCD data for the pion [30] and nucleon (LHPC [31] and QCDSF [32] collaborations), picking the valence quark contribution, are too noisy as to pin down the coupling of the excited scalar-isoscalar states to the energy-momentum tensor. Nevertheless, useful information confirming the (Regge) mass estimates for the a-meson can be extracted [16] through the use of the multiplicative QCD evolution of the GFF in the valence quark momentum fraction, (x)u+d, (23) (24) m^MeV] ttw [MeV] ttw [MeV] ttw [MeV] ttw [MeV] GFF GFF ( qq q ) 2 -dynam. (q q)2 -quench. 230 580(190) 620(100) - 400(30) 342 630(190) 660(90) - 720(20) 478 710(200) 730(90) - 1000(20) 318 620(190) 650(90) 468(50) - 469 700(190) 720(90) 936(13) - 526 739(200) 750(90) 1066(13) - Table 1. Scalar monopole mass obtained from the nucleon gravitational form factors, extracted from the (q q) components obtained by LHPC [31] and QCDSF [32] and compared to the lattice calculation using the tetraquark (q q)2 probing fields, both for the dynamical and quenched fermions [33] . as seen in deep inelastic scattering or on the lattice at the scale = 2 GeV. For the pion and nucleon GFFs we obtain the fits (27) yields mff = 550-180MeV and c = 0.95-0.70, or mff = 600-800MeV and c = 0.8(2), depending on the choice of the lattice data [31] or [32], respectively. Note that c is close to unity. Higher quark masses might possibly clarify whether or not the state evolves into a glueball or a meson. For a (qq)n system one expects mCT —> 2nmq + const at large current quark mass mq. The data from [31] or [32] are too noisy to see the difference, although for the largest pion masses we see that mCT ~ mn, as it simply corresponds to the q q-component of 0. We observe, however, that for mn ~ 500MeV our results are not far away from the recent lattice calculation using the tetraquark probes, (qq)2 [33], which provide mCT ~ 2mn for the largest pion masses as they should (see Table 1). From this viewpoint, and unless operator mixing is implemented, the nature of the state is predetermined by the probing operator. 8 Dimension-2 condensates One of the problems of the large-Nc Regge models [35] and their holographic relatives [36,37] is that they may contradict expectations from the OPE, as they involve dimension-2 condensates. For instance, the OPE for the n@@(q2) correlator in Eq.(12) gives corrections O(q0), while the O(q2) terms are missing [21]. This yields a one to one comparison: Co = - lim = -™LlL (ß^V (28) oo dm^/ dn 2n2 \ a J ' C2 = X fn = (29) ■ C4=If^ = (30) Equation (28) requires infinitely many states, while Eq. (29) suggests a positive and non-vanishing gauge-invariant dimension-2 object, C2 = i J" d4xx2(0(x)0), which is generally non-local, as it should not appear in the OPE. Note that C2 > 0, hence is non-vanishing for a finite number of states. The infinite Regge spectrum of Eq. (2) with Eq. (28) may be modeled with a constant ffo = fn,+ = O(Nc) whereas fn,_ = 0(\/ÑQ goes as Eq. (14) and yields a convergent and hence positive contribution. Naively, we get C2 = oo. However, C2 may vanish, as required by standard OPE, when infinitely many states are considered after regularization. The use of the Z-function regularization [16,34] gives C2 = limH f-riMs (n)2s =Y_ f£,- + ffo (1/2 - m20/a) . (31) Then C2 = 0 for mfo > ^fa/l = 810(40) MeV, a reasonable value to 0(1 /Nc) In any case, the important remark is that while at the OPE level C2 = 0 vanishes for trivial reasons, at the Regge spectrum level some fine tuning must be at work. Acknowledgments This work is partially supported by the Polish Ministry of Science and Higher Education (grants N N202 263438 and N N202 249235), Spanish DGI and FEDER funds (grant FIS2008-01143/FIS), Junta de Andalucía (grant FQM225-05). References 1. C. Amsler, et al., Phys. Lett. B667,1 (2008). 2. E. Witten, Nucl. Phys. B160, 57 (1979). 3. W. Broniowski, Nucl. Phys. A580, 429 (1994), hep-ph/9402206. 4. A. Pich (2002), hep-ph/0205030. 5. A. Andrianov, D. Espriu, and A. Prats, Int.J.Mod.Phys. A21, 3337 (2006), hep-th/0507212. 6. E. Gurvich, Phys.Lett. B87, 386 (1979). 7. A. Casher, H. Neuberger, and S. Nussinov, Phys.Rev. D20,179 (1979). 8. N. Suzuki, T. Sato, and T.-S. Lee, Phys.Rev. C79, 025205 (2009), 0806.2043. 9. R. Workman, R. 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Arriola, and W. Broniowski, Eur. Phys. J. A31, 739-741 (2007), hep-ph/0609266. 35. E. Ruiz Arriola, W. Broniowski, Phys. Rev. D73 (2006) 097502. 36. O. Andreev, Phys. Rev. D73 (2006) 107901. 37. F. Zuo, T. Huang, [arXiv:0801.1172 [hep-ph]]. Bled Workshops in Physics Vol. 12, No. 1 p. 18 Transversity structure of the pion in chiral quark models* W. Broniowskia,b, E. Ruiz Arriolac,d, A. E. Dorokhove,f a Institute of Nuclear Physics PAN, PL-31342 Cracow, Poland b Institute of Physics, Jan Kochanowski University PL-25406 Kielce, Poland c Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain d Instituto Carlos I de Fisica Teórica y Computacional, Universidad de Granada E-18071 Granada, Spain e Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics RU-141980 Dubna, Russia f Institute for Theoretical Problems of Microphysics, Moscow State University RU-119899 Moscow, Russia Abstract. We describe the chiral quark model evaluation of the transversity Generalized Parton Distributions (tGPDs) and related transversity form factors (tFFs) of the pion. The obtained tGPDs satisfy all necessary formal requirements, such as the proper support, normalization, and polynomiality. The lowest tFFs, after the necessary QCD evolution, compare favorably to the recent lattice QCD determination. Thus the transversity observables of the pion support once again the fact that the spontaneously broken chiral symmetry governs the structure of the Goldstone pion. The proper QCD evolution is crucial in these studies. This talk is based on our two recent works [1,2], where more details and a complete list of references may be found. Its topic concerns the transversity Generalized Parton Distribution (tGPD) of the pion, the least-known of the Generalized Parton Distributions (see [3-5] and references therein for an extensive review). The definition involves aligned parton-helicity operators (maximum-helicity case). For the case of spin-0 hadrons, tGPDs arise because of the nonzero orbital angular momentum between the initial and final state, thus allowing to study the spin structure without the inherent complications of the explicit spin degrees of freedom, as in the case of the nucleon. In that situation the analysis of the spin structure of the pion is particularly appealing, however, the quantity will be very difficult to access experimentally. A few years ago, however, lattice simulations [6] provided the lowest-order pion quark transversity form factors (tFFs), defined as Mellin moments of tGPDs * Talk delivered by W. Broniowski A20 Fig. 1. The quark part of the spin-2 gravitational form factor in Spectral Quark Model (solid line) and NJL model with the Pauli-Villars regularization (dashed line), compared to the lattice data from Ref. [7,8]. The band around the Spectral Quark Model results corresponds to the model parameter uncertainty. in the Bjorken x variable. That way lattices supply valuable information on the nontrivial spin structure of hadrons. In general, lattice calculations are capable to determine quantities that may only be dreamed off to be measured experimentally and, in that regard, are extremely useful. The results can be used to verify various theoretical approaches and models in their rich spectrum of predictions. An example is the gravitational form factor of the pion. Its lattice determination [7,8] agrees remarkably well with the evaluation in chiral quark models [9], as can be seen from Fig. 1. Our study consist of two distinct parts: 1) the chiral quark model determination of tFFs and tGPDs of the pion and 2) the QCD evolution. For the first part we apply the standard local NJL model with the Pauli-Villars regularization [10] and two versions of the nonlocal models, where the quark mass depends on the momentum of the quark, namely, the instanton model [11] and the Holdom-Terning-Verbeek (HTV) model [12]. We stress that chiral quark models have been successfully used for the evaluation of soft matrix elements entering numerous high-energy processes [9,13-40]. They also agree with the Euclidean lattice determination of moments (see, e.g., [41,42]) and direct results from the transverse lattices [43-46]. The second element, crucial in obtaining proper results, is the QCD evolution, where renorm-improved radiative gluonic corrections are appended. The method is schematically depicted in Fig. 2. One-loop (large-Nc) quark diagram, with external gauge bosons and Goldstone mesons, is evaluated. Then the renorm-improved gluon exchanges are incorporated in terms of the LO DGLAP evolution. The scale where the quark model calculation is carried out can be identified with the help of the momentum fraction carried by the quarks. According to phenomenology [47,48] or lattice calculations [49], the valence quark contribution is 47% of the total at the scale ^ = 2GeV. Since the quark models possess no explicit gluons, the valence quarks carry 100% of the momentum. This determines the quark model scale, denoted as as the scale determined in such a way, that Fig. 2. One-loop (large-Nc) quark diagram, with external gauge bosons (wavy lines) and Goldstone mesons (dashed lines). The renorm-improved gluon exchanges are incorporated in terms of the LO DGLAP evolution. when the evolution is carried out from |j.0 up to = 2 GeV, the fraction drops to 47 ± 2%. The result of the LO DGLAP evolution is Ho = 313+20 MeV. (1) Despite the low value of this scale, the prescription has been successfully confirmed by a variety of high-energy data and lattice calculations (see [26] and references therein). Moreover, the NLO DGLAP modifications yield moderate corrections [14], supporting our somewhat strained use of perturbative QCD at low scales. To summarize, our approach = chiral quark model + QCD evolution. We now come to definitions. The pion u-quark tFFs, denoted as Bnu (t), are defined via [50] (7t+(p,)|u(0)ia^aMK ^ u(0)|7t+(p)> = (a • P)^1 [a ' ^ ' P'] n-1 (2^)i Bnu (t) , (2) i=0, even where the auxiliary vectors a and b satisfy the conditions a2 = (ab) = 0 and b2 = 0. The skewness parameter is defined as £ = —a • q/(2a • P), the symbol ^_^ ^_^ ^_^ /_^ ^_ \ D P = 3 13 — igA13 is the covariant derivative, and = — 313]. Next, 2 p' and p are the initial and final pion momenta, P = ^(p' + p), q = p' — p, and 22 t = _q2. The bracket denotes antisymmetrization in the vectors a and b. The corresponding definition of the tGPD is [3] <7T+(p') | u(-a)i^a,bYu(a) | 7t+(p)) = dxe-ixP aEj,u(x, £,t), 1 (3) where the presence of the gauge link operator are understood. The tFFs for the d-quarks follow from the isospin symmetry, namely Bnd (t) = (_1 )n Bnu (t). The tFFs are the moments of the tGPD in the x-variable, n-1 dxxn-1E?>u (x,£,t) =Y (2£)i B^iu (t). (4) 1 i=o, even 1 7T Fig. 3. The form factors Bn0u(t) and Bj0u(t), evaluated at mK = 600MeV in the local NJL model (left panel) and in nonlocal models (right panel, solid line - HTV model, dashed line - instanton model). The lattice data from [6]. The local NJL and HTV models agree very well with the data. This formula explicitly displays the desired polynomiality property. We remark that the full information carried by tGPDs is contained in the collection of the infinitely many tFFs. The full details of the quark-model calculation as well as the QCD evolution can be found in [1,2]. The two lowest tFFs available from the lattice data, B^U and B^nU, evolve multiplicatively in a simple way: f «fuUYÎ/(2Pc) , (5) where are the appropriate anomalous dimensions [1,2]. In the local model, in the chiral limit at t = 0 we find the very simple result: R«.-ft n-„ 1/™ NcM B&"(t = 0;n) l(ccM\S/27 B10 (t^oVm^^, B«,u(t = 0;H)=3^J - where M is the constituent quark mass. The results of the model calculation followed by evolution are shown in Fig. 3. We note a striking agreement with the lattice data [6] for the local NJL model, as well as for the non-local HTV model. Finally, we present the results for the full tGPD for t = 0 and £ = 1/3 or £ = 0. The evolution is different for the symmetric and antisymmetric parts of tGPDs, hence we define the isovector and isoscalar combinations: E^1 (x, £, Q2) = E^5 (x, £, Q2) = E^ (x, £, Q2) + E^ (-x, £, Q2) , En'I=0 (x, £, Q2) = E?'a (x, £, Q2) = En (x, £, Q2) - E? (-x, £, Q2). The QCD evolution has been carried out with the method of [51-54]. The results for £ = 1/3 in the NJL model are shown in Fig. 4, while in Fig. 5 we compare the result for £ = 0 in the NJL model (left panel) and the nonlocal instanton model (right panel). Except for different end-point behavior, discussed in [2], the results are similar. In conclusion we wish to stress that the absolute predictions for the multi-plicatively evolved B10 and B20 agree remarkably well with the lattice results, X X Fig. 4. The symmetric (left panel) and antisymmetric (right panel) tGPDs of the pion at t = 0 and £, = 1/3, evaluated in the NJL model in the chiral limit at the quark-model scale |O = 313 MeV (solid lines) and evolved to the scales |i = 2 GeV (dashed lines) and 1 TeV (dotted lines). Fig. 5. The tGPD of the pion at t = 0 and £, = 0, evaluated in the chiral limit in the local NJL model (left panel) and in the instanton model (right panel). The solid lines correspond to the quark-model scale |0 = 313 MeV, the dashed lines to |i = 2 GeV, and the dotted lines to | = 1 TeV. supporting the assumptions of numerous other calculations following the same "chiral quark model + QCD evolution" scheme. Our study of the transversity observables of the pion support once again the feature that the spontaneously broken chiral symmetry determines the structure of the Goldstone pion. Acknowledgments This work is supported by the Bogoliubov-Infeld program (JINR), the Polish Ministry of Science and Higher Education, grants N N202 263438 and N N202 249235, Spanish DGI and FEDER grant FIS2008-01143/FIS, Junta de Andalucía grant FQM225-05, and EU Integrated Infrastructure Initiative Hadron Physics Project, contract RII3-CT-2004-506078. AED acknowledges partial support from the Russian Foundation for Basic Research, projects No. 10-02-00368 and No. 11-02-00112. References 1. W. Broniowski, A.E. Dorokhov and E.R. Arriola, Phys. Rev. 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D72 (2005) 114006, hep-ph/0509330. Bled Workshops in Physics Vol. 12, No. 1 p. 25 Spectroscopy of heavy baryons* Joseph P. Day, Ki-Seok Choi, Willibald Plessas Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria Abstract. We report first results from a study of heavy-baryon spectroscopy within a relativistic constituent-quark model whose hyperfine interaction is based on Goldstone-boson-exchange dynamics. 1 Introduction The relativistic constituent-quark model (RCQM) has become quite successful for the description of hadron properties at low energies. This is especially true for the RCQM based on Goldstone-boson-exchange (GBE) dynamics [1] with regard to baryons (for a short review see ref. [2]). So far the GBE RCQM has been restricted to baryons consisting of constituent quarks Q with flavors u, d, and s only, as it has been argued that their hyperfine interaction should be governed by GBE dynamics due to the spontaneous breaking of chiral symmetry (SBxS) of low-energy quantum chromodynamics (QCD) [3]. Regarding the other known baryons, i.e. the ones with flavors c and b, we still face the interesting questions after the light-heavy and heavy-heavy Q-Q interactions. It remains to be clarified, which kind of dynamics, gluon exchange and /or Goldstone-boson exchange, is dominant. We have looked into these problems within the framework of the RCQM. Accepting the GBE RCQM in the SU(3)F sector, there are in principle three ways to add interactions for the light-heavy and heavy-heavy Q-Q interactions: • employ GBE dynamics throughout, • extend the SU(3)F GBE RCQM with one-gluon exchange (OGE) for the c and b flavors, and • use a superposition of both the GBE and OGE hyperfine interactions beyond According to our present experience the best performance of a universal RCQM for all SU(5)f baryons is achieved by the first way [4]. Here, we thus report results of a SU(5)f RCQM that is based on GBE dynamics for baryons of all five quark flavors. * Talk delivered by J. Day SU(3)F. 2 Theory Our theoretical framework is relativistic quantum mechanics (RQM), which assumes a fixed number of relevant degrees of freedom and relies on an invariant mass operator M = Mfree + Mint fulfilling all symmetry requirements of the Poincare group. Here, the free and interaction parts of the mass operator are expressed in the rest frame of the baryon (i.e. M free X vm2+k2, Mint = x Vij = L 1+Vhf (i) ^<1 ^<1 where k represent the three-momenta of the individual quarks with rest masses mi and the Q-Q potentials Vj are composed of confinement and hyperfine interactions. By employing such a mass operator M2 = with baryon four-momentum = (ft, P), the Poincare algebra of all ten generators {ft, Pi, Ji, Ki}, for i = 1,2,3, [Pi,Pj]= 0, [Ki,ft] = iPi, [Ji, Pj] = ietjkPk, [Ji, A]= 0, ^ [Ji, îj] = ieijkîk, [Ki,Kj] = —ieijkJk, [Pi,A] = [Ji,Kj] = [Ki,Pj] 0, : ieijkKk, = i6ijA. is guaranteed. 3 3 3 3 The GBE RCQM з.1 The SU(3)f Sector The hyperfine interaction of the GBE RCQM for constituent quarks with flavors и, d, and s, confined by a linear potential Vi?nf(r) = Cr, reads V (r) V„(r) £ AtaAa + Vk (r) £ AtaAa + V (r)AtsAf + V ' (r) A°A° a = 1 a=4 CTi • CTi (2) where r is the relative vector between constituent quarks i and j. The A? represent the SU(3)f Gell-Mann matrices of flavor a and the CTi the SU(2) Pauli spin matrices of the individual constituent quarks. The GBE is described by the exchange of the octet of pseudoscalar mesons n, K, and n, where due to the U(1) anomaly also the singlet exchange n' is added. The corresponding regularized meson-exchange potentials, derived in instantaneous approximation, are expressed by Vy (r) 9Y 1 2n 12mi mi e-Hyr r -ayt AY Y = n,K,n,n ', (3) with gY the quark-meson coupling constant, |iy the exchanged meson mass, and AY a cut-off parameter. The complete parameterization of the GBE RCQM can be found in ref. [1]. An extended version of it, including beyond spin-spin forces also all other interaction components stemming from GBE dynamics, was published in ref. [5]. 3 7 e 2 M Y r 3.2 Generalization to SU(5)F In the spirit of the ansatz (3) we have generalized the GBE RCQM to SU(5)F in order to cover also heavy baryons, containing the flavors c and b, in a universal model. Keeping the confinement potential unaltered, the extended hyperfine interaction is proposed to be V hf(r) 3 7 ^ Vn(r) KK + VK(r) ^ W + V,8 (r)A?A? + -V,0 (r) + a=1 a=4 12 14 VD(r)^ AtaA,a + VDs(r) £ AtaA,a + V^(r^A?5 + a=9 a=13 19 21 23 VB(r) £ A?A? + Vbs (r) £ AtaA,a + Vbc(r) £ AtaA,a + a=16 a=20 a=22 Vn24 (r)A?4A2^ at • a, . It contains the GBE in SU(5)F, which is represented by the exchange of the 24-plet of pseudoscalar mesons plus the singlet n0. The various regularized mesonexchange potentials have the same functional dependence as in Eq. (3). The detailed parameterization is given in a forthcoming paper [6]. 3.3 Consistency of the Universal GBE RCQM Since SU(3) c SU(4) C SU(5), the generalized GBE RCQM should perform with similar or even better success as the corresponding SU(3)F model specifically for u-, d-, and s-flavor baryons. This is not immediately obvious, as the light-and strange-baryon sectors are now influenced by an altered singlet exchange, namely, no that corresponds to SU(5)F rather than to SU(3)F. In addition the exchanges of n15 and n24 come into play. We thus present in Figs. 1 and 2 first a comparison of the spectroscopy of light and strange baryons, as yielded by the original SU(3)F and the extended SU(5)f GBE RCQMs. As becomes clearly evident, the SU(5)F GBE RCQM performs equally well, in some instances even better, than the original SU(3)F one. In particular, the new model also produces the right level orderings in the N and A excitation spectra due to the specific flavor dependence in the hyperfine interaction in Eq. (4). 3.4 Results for Heavy-Baryon Spectra Next we present the predictions of the SU(5)F GBE RCQM for the spectra of c-and b-flavor baryons in comparison to experimental data available for states with at least 4- or 3-star status according to the PDG (see Fig. 3). It appears that all experimental results, for which also a definite JP is known, are reproduced quite well. M [MeV] 1800-_,_ 17""-_____ =_ I'.....- 150^ - ^^ ==a"" 1400-^^ 1300 - 1200 - 1100 - 1000 - 900 --- --- 1+ 1- 3- 5- 1- 3+ 3- 2 2 2 2 2 2 2 N A Fig. 1. N and A spectra of definite spin and parity JP produced by the extended SU(5)F GBE RCQM (left/red levels) in comparison to the ones of the original SU(3)F GBE RCQM [1,3] (right/blue levels) and to experimental data with their uncertainties (green boxes) from the PDG [7]. M [MeV] 1800- 1+ 1- 3- 1 + 3+ 3- 1+ 3+ 3- A E Fig. 2. Same as in Fig. 1 but for strange baryons. 3 + 2 Q 5 - 5 - 5 - 2 2 2 2 2 2 2 2 2 2 2 In Fig. 4 we also present the predictions of the SU(5)f GBE RCQM for double-charm baryons. Here, there is only one measurement reported by the PDG, namely, the ground state of ^cc. As can be seen from Fig. 4 and also the Table below, the theoretical level produced by the GBE RCQM remains at variance with the experimental data. For this comparison, however, one should bear in mind that the M [MeV] 3100- 3000 - 2900 - 2800 - 27002600 - 2500 -2400 - 2300 - 2200 1 + 2 3 -2 Ac 5 + 2 1 + 2 3 + 2 Sc 1 + 2 3 + 2 M [MeV] 6400 - 6300 - 6200 - 6100- 6000 - 5900 -5800 - 57005600 - 5500 - 1 + 2 Ab 1 + 2 3 + 2 Sb 1 + 2 fib Fig. 3. Heavy-baryon spectra of definite JP as produced by the extended SU(5)F GBE RCQM (solid / red levels) in comparison to experimental data with their uncertainties (dotted/green levels resp. boxes) reported by the PDG [7]. lowest Ecc state with Jp = j+ is only rated by 1 star by the PDG. Its measurement was only made once in 2002 by the SELEX collaboration [8], and since then has never been reproduced independently. In view of other theoretical works having investigated double-charm baryons, one may have some doubt about the measured mass of Ecc. As is evident from the comparison in the Table below, for instance, the theoretical results from the RCQM of the Bonn group [11] and also the ones from the Bhaduri-Cohler-Nogami one-gluon-exchange model [9], reported in 2005 by Stancu and Richard [10] at the Bled Workshop, give mass values for the Ecc ground state quite similar to the one we have achieved. Further measurements of double-charm baryons would thus be highly welcome. M [MeV] 440043004200410040003900 380037003600- 3500' 1 + 2 3 + 2 1 + 2 3 + 2 a Fig.4. Scc and flcc spectra as produced by the extended SU(5)f GBE RCQM (solid/red levels) in comparison to experimental data reported only for the Scc ground state (dotted/green level/box) [7,8]. Baryon JP Theory Experiment [8] Ref. [10] Ref. [11] GBE RCQM 3673 3518.9 ± 0.9 3711 - 3919 - 3919 - Table 1. Comparison of the predictions by the GBE RCQM and other theoretical models for double-charm Scc ground and excited states vis-a-vis the experimental measurement reported by the SELEX collaboration. 1 + 2 3 + 2 J_- 2 32 3643 3724 3642 3723 3920 3920 — cc —cc —cc -cc 4 Conclusion We have constructed a universal RCQM for all baryons with flavors u, d, s, c, and b. It is based on a relativistically invariant mass operator describing systems of three constituent quarks, confined by a linear potential according to QCD and interacting through hyperfine forces derived from GBE. This RCQM extends the previous GBE RCQM beyond SU(3)F and reproduces the phenomenologically known spectra with reasonable accuracy. For definitely pinning down the type of hyperfine interaction especially for light-heavy and heavy-heavy Q-Q subsystems more data in the sector of c- and b-flavor baryons would be highly desirable. In future it will be very interesting, if the universal GBE RCQM discussed here will be able to describe also reactions involving heavy baryons with a similar good performance as has previously been found for the SU(3) GBE RCQM in the cases of light and strange baryons. Acknowledgments This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16). J.P.D. would like to thank Profs. Ica Stancu and Veljko Dmitrasinovic for valuable discussions during the Workshop, giving him further insights into heavy-baryon spectroscopy. References 1. L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998). 2. W. Plessas, PoS LC2010, 017 (2010); arXiv:1011.0156 [hep-ph]. 3. L. Y. Glozman, Z. Papp, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. C 57, 3406 (1998). 4. J. P. Day, K. -S. Choi, and W. Plessas, arXiv:1108.3450 [hep-ph]. 5. K. Glantschnig, R. Kainhofer, W. Plessas, B. Sengl, and R. F. Wagenbrunn, Eur. Phys. J. A23, 507 (2005). 6. J. P. Day, K. -S. Choi, and W. Plessas, to be published. 7. K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). 8. M. Mattson et al. (SELEX Collaboration), Phys. Rev. Lett. 89,112001 (2002). 9. R. K. Bhaduri, L. E. Cohler, and Y. Nogami, Nuovo Cim. A65, 376 (1981). 10. J.-M. Richard and F. Stancu, in Exciting Hadrons (Proceedings of the Mini-Workshop, Bled, Slovenia, 2005), ed. by B. Golli, M. Rosina, and S. Sirca (DMFA, Zaloznistvo, Ljubljana, 2005), p. 25; [hep-ph/0511043]. 11. S. Migura, D. Merten, B. Metsch, and H. -R. Petry, Eur. Phys. J. A28, 41 (2006). Bled Workshops in Physics Vol. 12, No. 1 p. 32 Baryons' anomalous magnetic moments in a Ul(3) X Ur(3) chiral symmetric theory* V. Dmitrasinovica, Hua-Xing Chenb,c, Atsushi Hosakad a Institute of Physics, Belgrade University, Pregrevica 118, Zemun, P.O.Box 57,11080 Beograd, Serbia b School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China c Departamento de Física Teorica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigation de Paterna, Aptdo. 22085, 46071 Valencia, Spain d Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047, Japan Three-quark nucleon interpolating fields in QCD have well-defined SUL(3) x SUr(3) and UA(1) chiral transformation properties, viz. [(6,3) © (3,6)], [(3,3) © (3,3)], [(8,1) © (1,8)] and their "mirror" images, Ref. [13]. It is known, Ref. [3] that chiral mixing of the [(6,3) © (3,6)] multiplet with one ordinary ("naive") and one "mirror" field belonging to the [(3,3) © (3,3)] and [(1,8) © (8,1)] multiplets allows fitting of the isovector (gA) and the flavor-singlet (isoscalar) axial coupling (gA) of the nucleon. The magnetic moments of baryons are, however, difficult to incorporate into such a chiral mixing scheme. In order to reproduce the anomalous magnetic moments of baryons, we construct all SUL(3) x SUR(3) chirally invariant one-derivative Pauli tensor one-[(8,1) © (1,8)]-meson-baryon interactions subject to chiral mixing. It turns out that there are (strong) selection rules: for example, there is only one one-derivative chirally symmetric interaction between J = \ fields belonging to the [(6,3) © (3,6)] and the [(3,3) © (3,3)] chiral multiplets. We also study the chiral anomalous magnetic interactions of the [(3,3) © (3,3)] and [(8,1) © (1,8)] baryon fields. Again, there are selection rules that allow only off-diagonal and no diagonal one-derivative chiral SUL (3) x SUR(3) interactions of this type, that also conserve the UA (1) symmetry. We use these interactions to calculate the F/D ratio for the anomalous magnetic moments of baryons as F/D=1/3, in close proximity to the experiment. In this talk we report some as yet unpublished results [1] of our studies of the anomalous magnetic moments of the baryons in the chiral mixing approach. Recent studies [2,3] point towards baryon chiral mixing (of [(6,3) © (3,6)] with the [(3,3) © (3,3)], [(8,1) © (1,8)] chiral multiplets as a possible mechanism underlying the baryons' axial couplings. This finding is in line with the old current algebra results of Gerstein and Lee [4,5] and of Harari [6,7], updated to include * Talk delivered by V. Dmitrasinovic 1 These multiplets are not limited to three-quark interpolators. recently measured values of F and D couplings, Ref. [8], and extended to include the flavor-singlet coupling g A of the nucleon [9,10], which was not considered in the mid-1960's at all, presumably due to the lack of data. Our own starting point was the QCD interpolating fields' UA(1) chiral properties [11-13]. Even though the chiral mixing has been known for more than 40 years [1417], the SUl(3) x SUR (3) chiral interactions necessary to describe the anomalous magnetic moments have not been discussed in print, only the problems associated with them [18]. It ought to be noted, however, that Gerstein and Lee [5] found a phenomenological chiral mixing scheme that led to acceptable anomalous magnetic moments of the nucleons. These authors did not try to construct a chiral Lagrangian that would reproduce such chiral mixing, however. The present paper serves to provide a dynamical model of chiral mixing that is the "best" approximation to the phenomenological solution of both the (F, D) and the flavor-singlet axial coupling, and the anomalous magnetic moment problems, assuming only three-quark baryon interpolating fields. In our previous publication [19] we found two solutions that fit the axial coupling data 2: one that conserves the UA(1) symmetry (the Harari scenario) and another one that does not (the Gerstein-Lee scenario). Here we have shown that only the former scenario leads to nucleon anomalous magnetic moments that are in agreement with experiment. Having made the first step, which was to reproduce the phenomenologi-cal mixing starting from a chiral effective model interaction, we turn to the next step, which is to look for a chiral dynamical source of the anomalous magnetic moments. One such mechanism is the simplest chirally symmetric one-derivative one-(p, a)-meson interaction Lagrangian; one-derivative because only thus one can couple the baryon magnetic moment (the Pauli current) to the p-field. Here we study vector meson couplings because photon couplings follow them under the vector meson dominance hypothesis which has been shown to work in the low energy region. We have constructed all SUL(3) x SUR(3) chirally invariant one-derivative one-vector-meson-baryon interactions and then use them to calculate the baryons' magnetic moments as well as the non-derivative Dirac terms. There are severe chiral symmetry induced selection rules, see Table 1. We used these interactions to relate the anomalous magnetic moments to the "physical" mixing angles determined from the axial couplings and the baryons' masses. In this way we found a unique solution that unequivocally points towards the Harari scenario as the phenomenologically correct one. Of course, the absolute values of the anomalous magnetic moments cannot be determined by mere use of the chiral symmetry, without dynamical caculations, but their F/D ratio can. The magnetic moment F/D ratio is predicted to be 1/3 by the chiral mixing interaction, the same value as in the non-relativistic quark model, or as in the SU(6) symmetry limit. This last fact is very curious and seems to require further thought. 2 this does not preclude the existence of more complicated solutions. Table 1. Allowed chiral invariant Pauli type interaction terms with one (8,1) © (1,8) vector meson field N 3VM^N. In the first column we show the chiral representation of N, and the first row the chiral representation of N. We use "[mir]" to denote the relevant mirror fields. (8,1) ©(1,8) (3,3) ©(3,3) (6,3) © (3,6) (10,1) ©(1,10) (1,8) ©(8,l)[mir] 2 xMn Mn (3,3)©(3,3)[mir] Ml (3,6) © (6,3)[mir] Ml (1,10) ©(10,1) [mir] Mn Mn (3,3) ©(3, 3) (6,3) © (3,6) (3,3) © (3,3) Mn (6,3) © (3,6) M* The "Gerstein-Lee" scenario requires vanishing nucleon anomalous magnetic moments, in serious disagreement with experiment (here we ignore any and all chiral mixing in the vector meson sector, which also violates the UA(1 )-symmetry [20]). This goes to show that the "QCD UA(1) anomaly" probably does not play a role in the "nucleon spin problem" [9,10], as was once widely thought [21]. In all likelihood the UA(1) anomaly provides only a (relatively) small part of the solution, associated with the higher Fock space components, whereas the largest part comes from the UA (1 )-symmetric chiral structure ("mixing") of the three-quark components of the nucleon. The next step, left for the future, is to investigate the SU(3) x SU(3) —> SU(2) x SU(2) symmetry breaking and the study of the chiral SU(2) x SU(2) properties of hyperons. Then one may consider explicit chiral symmetry breaking corrections to the axial and the vector currents, which are related to the SU(3) x SU(3) symmetry breaking meson-nucleon derivative interactions, not just the explicit SU(3) symmetry breaking ones that have been considered thus far (see Ref. [8] and the previous subsection, above). References 1. H. X. Chen, V. Dmitrasinovic and A. Hosaka, submitted to Phys. Rev. D (2011). 2. V. Dmitrasinovic, A. Hosaka and K. Nagata, Mod. Phys. Lett. A 25, 233 (2010). 3. H. X. Chen, V. Dmitrasinovic and A. Hosaka, Phys. Rev. D 81, 054002 (2010). 4. I. S. Gerstein and B. W. Lee, Phys. Rev. Lett. 16,1060 (1966). 5. I. S. Gerstein and B. W. Lee, Phys. Rev. 152,1418 (1966). 6. H. Harari, Phys. Rev. Lett. 16, 964 (1966). 7. H. Harari, Phys. Rev. Lett. 17, 56 (1966). 8. T. Yamanishi, Phys. Rev. D 76, 014006 (2007). 9. W. Vogelsang, J. Phys. G 34, S149 (2007). 10. S. D. Bass, "The Spin structure of the proton," World Scientific, Singapore, (2007). 11. K. Nagata, A. Hosaka and V. Dmitrasinovic, Mod. Phys. Lett. A 23, 2381 (2008). 12. K. Nagata, A. Hosaka and V. Dmitrasinovic, Eur. Phys. J. C 57, 557 (2008). 13. H. X. Chen, V. Dmitrasinovic, A. Hosaka, K. Nagata and S. L. Zhu, Phys. Rev. D 78, 054021 (2008). 14. Y. Hara, Phys. Rev. 139, B134 (1965). 15. W. A. Bardeen and B. W. Lee, Phys. Rev. 177, 2389 (1969). 16. B. W. Lee, Phys. Rev. 170,1359 (1968). 17. S. Weinberg, Phys. Rev. 177, 2604 (1969). 18. R. Dashen and M. Gell-Mann, Phys. Rev. Lett. 17, 340 (1966). 19. H. X. Chen, V. Dmitrasinovic and A. Hosaka, Phys. Rev. D 83, 014015 (2011). 20. V. Dmitrasinovic, Nucl. Phys. A 686, 379 (2001). 21. H. q. Zheng,"Singlet axial coupling, proton structure and the parity doublet model," CERN-TH-6327-91 preprint (unpublished). Bled Workshops in Physics Vol. 12, No. 1 p. 36 On the way to a realistic description of hadron resonances* Regina Kleinhappel, Willibald Plessas, Wolfgang Schweiger Theoretical Physics, Institute of Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria We are in the course of developing a coupled-channel relativistic constituent-quark model (CC RCQM). Thereby it should become possible to describe hadron reactions more realistically. In particular, we focus on strong hadron resonance decays, where we want to include the coupling to the mesonic decay channels explicitly. In this regard, promising results have already been obtained before in a toy model for quark-antiquark systems with a scalar interaction neglecting spin and flavor degrees of freedom [1,2]. There we calculated the decay of a meson resonance into the ground state by emitting a pion in a fully relativistic manner. The RCQM was constructed in a coupled-channel formalism along Poincare-invariant quantum mechanics in the point-form. It leads to the interacting mass operator in matrix form comprising the two channels 1. the confined quark-antiquark system, depending on the valence-quark degrees of freedom, described by the mass operator Aval and 2. the decay channel, containing in addition the n as the decay product, described by the mass operator KXval>n: M=(ML. )• (1) Here, the operator K provides the coupling to the decay channel by producing the n at an elementary quark/antiquark-n vertex. For simplicity, the mass operator in the first channel is assumed to be the free mass operator plus a confinement interaction of harmonic oscillator type. The mass operator Mval,n in the second channel contains in addition the kinetic energy of the n. We solved the eigenvalue problem of the matrix mass operator in Eq. (1) after a Feshbach reduction leading to the complex eigenvalue problem Aval + K + (M - Aval,« + i0)-1 K] |^val) = M l^val) • (2) The results for the mass eigenvalues are shown in Fig. 1 as a function of the coupling strength to the decay channel. For the definite value of g2/4n=1.19, marked * Talk delivered by R. Kleinhappel by the vertical line, we obtained the ground state corresponding to a model p meson with real mass eigenvalue m and the first excitation as a true resonance with complex mass eigenvalue M. The decay width of the latter is r = 2 Im M = 26 MeV. 4 n Fig. 1. Dependence of the mass-operator eigenvalues of the ground state (lower solid curve) and the first excited state (upper solid curve) on the coupling constant between the constituent (anti)quarks and the meson. For the resonant state it is also shown, how the decay width F - whose value is multiplied by a factor of 4 for better visibility - develops (shaded area). The vertical line indicates the particular coupling strength, where the mass eigenvalues of a model p ground state and an w* resonance are reproduced. The horizontal line marks the energy, where the decay width F vanishes; it is just the energy of the Q Q ground state with only confinement plus the n mass. Beyond describing the spectrum more realistically, our model also allows to deduce meson vertex form factors from a microscopic approach. This has been done in ref. [3]. Currently, we are applying our approach to mesons, including spin and flavor degrees of freedom. At the same time we are improving the dynamics entering the valence and decay channels. Subsequently, the whole formalism will be extended to baryons. We expect that the notorious shortcomings of singlechannel models, producing hadronic decay widths generally too small [4], will thereby be remedied. Acknowledgments This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16). References 1. R. Kleinhappel and W. Schweiger, in Dressing Hadrons (Proceedings of the MiniWorkshop, Bled, Slovenia, 2010), ed. by B. Golli, M. Rosina, and S. Sirca, DMFA, Ljubljana (2010), p. 33; arXiv:1010.3919 2. R. Kleinhappel and W. Schweiger, to appear in the Proceedings of the 14th International Conference on Hadron Spectroscopy, Munich, 2011; arXiv:1109.0127 3. R. Kleinhappel, Diploma Thesis, University of Graz, 2010 4. T. Melde, W. Plessas, and B. Sengl, Phys. Rev. D 77,114002 (2008) Bled Workshops in Physics Vol. 12, No. 1 p. 39 The Schwinger model in point form* D. Kupelwiesera, W. Schweigera, and W. H. Klinkb a Institut fur Physik, Universitat Graz, A-8010 Graz, Austria b Dept. Physics and Astronomy, The University of Iowa, Iowa City, IA 52242-1479, U.S. Abstract. We attempt to solve the Schwinger model, i.e. massless QED in 1+1 dimensions, by quantizing it on a space-time hyperboloid x^x^ = t2 . The Fock-space representation of the 2-momentum operator is derived and its algebraic structure is analyzed. We briefly outline a solution strategy. 1 Introduction The Schwinger model is quantum electrodynamics of massless fermions in 1 space and 1 time dimension [1] and serves as a popular testing ground for non-perturbative methods in quantum field theory (QFT). It is an exactly solvable, super-renormalizable gauge theory that exhibits various interesting phenomena [2], such as confinement, which one would like to understand better in 1+3-dimensional QFTs. Originally it was solved by means of functional methods [1]. Later on also operator solutions were found [3] and spectrum and eigenstates of the theory were calculated by quantizing it at equal time x0 = const. [4,5] or at equal light-cone time x+ = x0 + x1 = const. [6]. We rather attempt to solve the Schwinger model by means of canonical quantization on the space-time hyperboloid xo xi — t 2. Each of these quantization hypersurfaces is associated with a particular form of relativistic Hamiltonian dynamics [7], namely the instant form, the front form and the point form, respectively. The quantization surface in point form is a space-time hyperboloid which is invariant under the action of the Lorentz group. The kinematic (interaction independent) generators of the Poincare group are therefore those of the Lorentz subgroup. All the interactions go into the components of the 2-momentum PH, i.e. the generators of space-time translations, which provide the dynamics of the system. One of the main virtues of point-form dynamics is obviously a simple behavior of wave functions and operators under Lorentz transformations. This has already been exploited in applications to relativistic few-body systems [8], but corresponding studies of interacting quantum field theories are still very sparse. The best-known paper is that of Fubini et al. [9], who deal with point-form QFT in 2-dimensional Euclidean space-time. We rather want to extend equal-T quantization in Minkowski space-time, as it was worked out in Ref. [10] for free field theories, to the interacting case. The solution being known, the Schwinger model * Talk delivered by D. Kupelwieser would be an interesting example to test the point-form approach against other methods. The hope is then that point-form quantum field theory will eventually represent a useful alternative in the study of 4-dimensional quantum field theories. The Lagrangian of the Schwinger model is 1 i ^ 1 photon part fermion part interaction part £ = £Y + £e + £int = --F^ + 0 (1) with the 2 x 2 Dirac matrices being represented, as usual, in the Weyl basis, i.e. Y° = o"i , y1 = ia2 and y5 = Y°Y1 = — °3 . 2 The 2-Momentum Operator 2.1 The free part This exposition follows closely Ref. [10] to which we refer for further details. Fermions: In order to obtain the Fock-space representation of the free fermion 2-momentum operator, we Fourier-expand the Dirac field ^ (x) in terms of plane waves using the fermion and antifermion annihilation (creation) operators c^ (p) and d(t) (p) and the spinor basis {u(p), v(p)}. In the massless case, the spinors are (P° = |p1l): u(p) 1 and v(p)=^fP;-p°V (2) \/2p° VP + P / v/VVP +v°J The free fermion 2-momentum operator in point-form is then obtained from the stress-energy tensor by integrating over the space-time hyperboloid = t2: P^ 2d2x6(x2-t2) 9(x°)xv with = (3) point-form "surface" element Inserting now the plain-wave expansion for the fields and interchanging momentum and x integrations we are left with the covariant distribution Wv(q) = 2 d2x 6(x2 - t2) 9(x0) xv e-iqx = 27t6(q2)e(q°)qv +27te(q2)6(q0)Io(TV/q2)g.o nT 2iT 9(q2) [iYi (xv^q2) + e(q°)Ji (xv^q2) F9(-q2)K1(tV/^q2)q.. (4) When evaluating equation (3) for the free parts of the Lagrangian (1), Wv is contracted with spinor products of the form Uyvu, Uyvv, etc. All the contractions q V with qv vanish and only the term N(Q2) = GMN(Q2) = 3 1 GMp(Q2) nu;N 2M -3 1 nd,N 2M 3 1 / nu;N Q \ -3 1 A nd,N Q V ', M V', M V ', M 2 2 2 ' 2 2 ' 2 2 ' 2 ' 2 ' 2 V, M 11 V, M Ji > d V, M, 2' 2 2' 2 2 ' 2 ' '2' 2 (9) (10) (11) (12) where J0'1 are the projections of the current-operator components J0,1 on flavors f = u, d and nf,N are the numbers of quarks with flavors u and d in the proton or the neutron, respectively. A priori we thus have eight separate contributions to the full results of nucleon Sachs form factors. Under charge symmetry they are reduced to only four independent contributions, since we have the restricting relations for both the electric and magnetic form factors GU;M (Q2)= 2Gd,M(Q2):= GU,M(Q2) 2Gd'M(Q2) GE,M (Q2):= G d GE,M (Q2) (13) (14) This means that the Sachs form factors of the proton and neutron are constituted from their flavor parts as 2 G? = =-G Gp 3 -G M Igî -G M Gn rn GM 2 -G 1 M pu (15) (16) The inverse formulae read GJU = 2GE + Gn , e gm = 2gm + g n M GE = GE + 2G£ GM = GM +2G n M (17) (18) U U U E d d U The corresponding flavor contributions to the nucleon Dirac and Pauli form factors are then given by ^Î(Q2) = T^[GE(Q2)+^GfM(Q2)] (19) ^(Q2) = [GfM(Q2) - G^(Q2)] , (20) for f=u, d. The latter can also be expressed directly through the proton and neutron Dirac and Pauli form factors in the following way Fu = 2Fp + Fn , Fd = 2Fn + Fp , i = 1,2. (21) 3 Results Here we present results for the flavor contributions to the nucleon Sachs as well as Dirac and Pauli form factors obtained from the GBE RCQM in the PFSM approach, i.e. through calculations in complete analogy to the previous studies of the nucleon electromagnetic structures [4-6,9]. The individual flavor contributions to the proton and neutron electric as well as magnetic form factors, as published in the original refs. [4,5], are shown in Fig. 1. As is nicely seen, not only the global results for all four form factors G]E, GM, G]n, and GM agree well with the world experimental data but also the separate flavor parts match the data extracted in ref. [1]. Slight differences to these data are only visible for the anyway very small d-flavor contribution to the magnetic form factor of the neutron. In addition to the electromagnetic Sachs form factors as depicted in Fig. 1 we may also examine the flavor compositions of the nucleon electric radii rE and magnetic moments |x. We recall that these quantities are defined through the behaviours of the electric and magnetic form factors at Q2=0 in the following way -dGE(Q2) -6- dQ2 , M = Gm(Q2 = 0). (22) Q2 =0 In Tables 1 and 2 we present the individual flavor contributions to the nucleon electric radii and magnetic moments. It is interesting how the different parts contribute to build up the full results, which are again in quite reasonable agreement with experimental data. The ratios of the Pauli to the Dirac form factors F2/Fi for the proton and neutron have been in the focus of many theoretical investigations because of scaling considerations. In ref. [1] the particular Q2 dependences have been highlighted in comparison to the corresponding ratios for the separate flavor parts, i.e. F2/F{, for f=u, d. From the flavor decomposition of the experimental data it was found that both FU/FU and Fd/Ff become practically constant for momentum transfers Q2 >1.5 GeV2, much in contrast to F2/Fi for both the proton and the neutron. We depict the behaviour of the theoretical results obtained with the GBE RCQM in Figs. 2 and 3. Again it is found that the theoretical predictions are in reasonable agreement with the experimental data not only for the total results (Fig. 2) but also for the flavor-separated ratios (Fig. 3). E gm(q2) - 2/3 Gm(Q2) - -1/3 GM(Q2) - GM(Q2) Bartel Hoehler Christy Sill Walker Andivahis Quattan CJRW: 2/3 GMu(Q2) CJRW: -1/3 GMd(Q2) Gao Xu 2003 Xu 2000 Bruins CJRW: 2/3 GM(Q2) CJRW: -1/3 G^(Q2) Fig. 1. u- and d-flavor contributions to the proton (upper panels) and neutron (lower panels) electric and magnetic form factors as predicted by the GBE RCQM in comparison to experimental data from ref. [1] (CJRW) and other experiments as indicated. Table 1. u- and d-flavor contributions to the proton and neutron electric radii squared rf [fm2 ]. GBE RCQM Flavor Contributions Total Experiment u d p 1.0089 -0.1848 0.8241 0.769(28) [12] 0.70869(113) [13] n 0.3696 -0.5045 -0.1349 -0.1161(22) [12] Table 2. u- and d-flavor contributions to the proton and neutron magnetic moments |i [n.m.]. GBE RCQM Flavor Contributions Total d Experiment u P n 2.4641 0.2343 2.6984 -0.4686 -1.2321 -1.7006 2.792847356(23) [12] -1.9130427(5) [12] 1 2 ,2w_2. -.2 , ,2., 2, Q- [(GeV")/c"] C [(Gev )/c'] 2w.2 2w_2 Q [(GeV)/cn Q" [(GeV")/c"] Fig. 2. Ratios of Pauli to Dirac form factors for the proton (left) and neutron (right) as predicted by the GBE RCQM in comparison to experimental data from ref. [1] and other experiments as indicated. 1 p—,—,—,—,—,—,—,—,—,—,—,—,—,—,—,—,—_ 0.5 - ^ —¿—Sr------------&_______________J -----f2/FU (Q2) o------Fd/Fd (Q2) : a CJRW: Fjj/F" (Q2) -05 . v CJRW: Ff/F? (Q2) : -1 " F2/F? (Q2) ; -1.5 2 ^-w -------■ - *" j -3 -,-,-,-,-,-,-,--,-,-,-,-,-,-,-L 0 12 3 Q2 [(GeV2)/c2] 2 -----FU/F1 (Q2) -----Fd/Fl (Q2) CJRW: Fj/Flj (Q2) CJRW: F^/F, (Q2) t t Fq/FÎ (Q2) -f—— Fig. 3. Ratios of the u- and d-flavor contributions to the Pauli to Dirac form factors as predicted by the GBE RCQM in comparison to experimental data from ref. [1]. 4 Conclusions From the flavor analysis of elastic electromagnetic nucleon form factors we have learned that the RCQM, especially the one with GBE hyperfine forces, provides rather reasonable results in practically all instances. We attribute the successes mainly to: • appropriate dynamics in the GBE RCQM, • very precise nucleon wave functions from the GBE RCQM, • mixed-symmetry spatial components in the nucleon wave functions, • manifest covariance of the observables due to point-form Lorentz boosts, • strict frame independence of the construction of the current operator, • fulfillment of current conservation. It is interesting to note that the GBE RCQM relies on three-quark configurations only. The underlying dynamics are just based on coupling valence-quark fields with Goldstone bosons. No explicit mesonic effects (e.g., so-called meson-cloud effects) or more-quark components are introduced beyond. From our previous studies we do know that relativistic (boost) effects are most important in the reproduction of the nucleon electromagnetic form factors [4,5]. This is even true for the quantities extracted at or near zero momentum transfers, i.e. the magnetic moments and electric radii [6]. For the good performance of the GBE RCQM we identify as most important two special symmetry ingredients in our theory, namely, the spontaneous breaking of chiral symmetry of low-energy QCD and strict Lorentz invariance. Acknowledgments This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16). The authors are grateful to B. Wojtsekhowski for providing them with an extensive experimental data base of elastic nucleon electromagnetic form factors. M.R. thanks the organizers of the Workshop for the generous support enabling him to participate in an exciting meeting with a lot of illuminating discussions. References 1. G. D. Cates, C. W. de Jager, S. Riordan, and B. Wojtsekhowski, Phys. Rev. Lett. 106, 252003 (2011) 2. L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998) 3. L. Y. Glozman, Z. Papp, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. C 57, 3406 (1998) 4. R. F. Wagenbrunn, S. Boffi, W. Klink, W. Plessas, and M. Radici, Phys. Lett. B511, 33 (2001) 5. S. Boffi, L. Glozman, W. Klink, W. Plessas, M. Radici, and R. Wagenbrunn, Eur. Phys. J. A14,17(2002) 6. K. Berger, R. F. Wagenbrunn, and W. Plessas, Phys. Rev. D 70, 094027 (2004) 7. P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949) 8. T. Melde, L. Canton, W. Plessas, and R. F. Wagenbrunn, Eur. Phys. J. A25, 97 (2005) 9. T. Melde, K. Berger, L. Canton, W. Plessas, and R. F. Wagenbrunn, Phys. Rev. D 76, 074020 (2007) 10. M. Rohrmoser, K. -S. Choi, W. Plessas, arXiv:1110.3665 [hep-ph] 11. W. Plessas, PoS LC2010, 017 (2010); arXiv:1011.0156 [hep-ph] 12. K. Nakamura et al. (Particle Data Group), J. Phys. G37, 075021 (2010) 13. R. Pohl et al., Nature 466, 213 (2010) Bled Workshops in Physics Vol. 12, No. 1 p. 55 A review of new data on compact stars from a Magnetar model of magnetised cores* Vikram Sonia and N. D. Haridassb a National Physical Laboratory and Jamia Millia University, New Delhi, India b Indian Insitute of Science, Bangalore, India Abstract. We have used the Magnetar model to identify some stars, in a sampling of a few high magnetic field pulsars, as magnetars. Thus this model throws up a lot of unexplored physics from the strongly interacting core to the plasma physics and the crustal solid state physics of huge magnetic fields. Our understanding of neutron stars is at a crossroad. We have to understand many families of neutron stars, for example pulsars and magnetars, in one framework. This is what we have tried to do in this work. Neutron stars are also the laboratory to understand the high density phase diagram for strong interactions. This work gives us a new understanding of the strong interactions that is linked intimately to astrophysical data. Our Model Conclusions In conclusion we enumerate some of the consequences of the model presented above: i) Magnetars belong exclusively to the higher than pulsar mass population of neutron stars that are born with a high density magnetic core. ii) The high density core is created by a strong interaction phase transition that aligns magnetic moments to create large dynamical 1016(17) G magnetic fields at the surface of the core. Dynamical fields are 'permanent' unlike fields derived from currents. iii) The core field is shielded by Lenz currents generated in the high conductivity plasma in and around it, but is gradually transported to the crust by ambipolar diffusion over a timescale of ~ 105-6 years with interior temperatures of more than ~ 108 5 K, - this results in a time delay before the field comes out to the surface. In old spun up binary neutron stars created by slow accretion (for example the large mass ( almost 2 solar mass ) binary neutron star, PSR J1614-2230), the lower interior temperatures inhibit transport of the core field to the crust. iv) The strong magnetic field breaks through the crust as the shielding currents dissipate giving out a steady X-ray flux and several energetic flares. v) This further implies that the surface field keeps increasing in magnitude till all shielding currents dissipate and the permanent dipolar core field is established. * Talk delivered by V. Soni vi) Neutron stars are also the laboratory to understand the high density phase diagram for strong interactions. The existence of the large mass ( almost 2 solar mass ) binary neutron star, PSR J1614-2230, very probably rules out soft equations of state, associated with quark matter (with/without condensate) cores. We have found that all these phenomena are supported by extensive data and observations. References 1. V. Soni and D. Bhattacharya, Phys. Lett. B 643 2006. 2. D. Bhattacharya and V. Soni, arXiv (astro-ph) 0705.0592, 2007. 3. N. D. Hari Dass and V. Soni, Magnetars from Magnetized Cores created by a Strong Interaction Phase Transition, arXiv(astro-ph) 1012.1420, 2010. Bled Workshops in Physics Vol. 12, No. 1 p. 57 Negative parity nonstrange baryons in large N c QCD: quark excitation versus meson-nucleon scattering* N. Matagnea and Fl. Stancub a Service de Physique Nucleaire et Subnucleaire, University of Mons, Place du Parc, B-7000 Mons, Belgium b Institute of Physics, B5, University of Liege, Sart Tilman, B-4000 Liege 1, Belgium Abstract. We show that the two complementary pictures of large N c baryons - the singlequark orbital excitation about a symmetric core and the meson-nucleon resonance - are compatible for I = 3 SU(4) baryons. The proof is based on a simple Hamiltonian including operators up to order O(N0) used previously in the literature for I = 1. 1 The status of the 1 /N c expansion method The large Nc QCD, or alternatively the 1/Nc expansion method, proposed by 't Hooft [1] and implemented by Witten [2] became a valuable tool to study baryon properties in terms of the parameter 1 /Nc where Nc is the number of colors. According to Witten's intuitive picture, a baryon containing Nc quarks is seen as a bound state in an average self-consistent potential of a Hartree type and the corrections to the Hartree approximation are of order 1/Nc. Ten years after 't Hooft's work, Gervais and Sakita [3] and independently Dashen and Manohar in 1993 [4] derived a set of consistency conditions for the pion-baryon coupling constants which imply that the large Nc limit of QCD has an exact contracted SU(2Nf)c symmetry when Nc —> oo, Nf being the number of flavors. For ground state baryons the SU(2Nf) symmetry is broken by corrections proportional to 1/Nc [5,6]. Analogous to s-wave baryons, consistency conditions which constrain the strong couplings of excited baryons to pions were derived in Ref. [7]. These consistency conditions predict the equality between pion couplings to excited states and pion couplings to s-wave baryons. These predictions are consistent with the nonrelativistic quark model. A few years later, in the spirit of the Hartree approximation a procedure for constructing large Nc baryon wave functions with mixed symmetric spin-flavor parts has been proposed [8] and an operator analysis was performed for I = 1 baryons [9]. It was proven that, for such states, the SU(2Nf) breaking occurs at order N0, instead of 1/Nc, as it is the case for ground and also symmetric excited states [56,1+] (for the latter see Refs. [10,11]). This procedure has been * Talk delivered by I. Stancu extended to positive parity nonstrange baryons belonging to the [70, 1+] with I = 0 and 2 [12]. In addition, in Ref. [12], the dependence of the contribution of the linear term in Nc, of the spin-orbit and of the spin-spin terms in the mass formula was presented as a function of the excitation energy or alternatively in terms of the band number N. Based on this analysis an impressive global compatibility between the 1 /Nc expansion and the quark model results for N = 0,1,2 and 4 [13] was found (for a review see Ref. [14]). More recently the [70,1-] multiplet was reanalyzed by using an exact wave function, instead of the Hartree-type wave function, which allowed to keep control of the Pauli principle at any stage of the calculations [15]. The novelty was that the isospin-isospin term, neglected previously [9] becomes as dominant in A resonances as the spin-spin term in N* resonances. The purpose of this work is to analyze the compatibility between the 1 /Nc expansion method in the so-called quark-shell picture and the resonance or scattering picture defined in the framework of chiral soliton models. Details can be found in Ref. [16]. 2 Negative parity baryons If an excited baryon belongs to a symmetric [56]-plet the three-quark system can be treated similarly to the ground state in the flavour-spin degrees of freedom, but one has to take into account the presence of an orbital excitation in the space part of the wave function [10,11]. If the baryon state is described by a mixed symmetric representation, [70] in SU(6) notation, the treatment becomes more complicated. In particular, the resonances up to 2 GeV belong to [70,1-], [70,0+] or [70,2+] multiplets and beyond to 2 GeV to [70,3-], [70,5-], etc. In the following we adopt the standard way to study the [70]-plets which, as already mentioned, is related to the Hartree approximation [8]. An excited baryon is described by a symmetric core plus an excited quark coupled to this core, see e.g. [9,12,17,18]. The core is treated in a way similar to that of the ground state. In this method each SU(2Nf) x O(3) generator is separated into two parts Si = si + Sc; Ta = ta + Tca; Gia = gia + Gca; li = lTq + 4, (1) where si, ta, gia and iq are the excited quark operators and Sc, T?, Gca and l£ the corresponding core operators. 2.1 The quark-shell picture In the quark-shell picture we use the procedure of Ref. [19], equivalent to that of Ref. [20], later extended in Ref. [21]. We start from the leading-order Hamiltonian including operators up to order O(N0) which has the following form H = c11 + c2£-s + C3^-£,2) -g-Gc (2) This operator is defined in the spirit of a Hartree picture (mean field) where the matrix elements of the first term are proportional to Nc on all baryons [2]. The spin-orbit term l • s which is a one-body operator and the third term - a two-body operator containing the tensor l(2)lj of O(3) - have matrix elements of order O(N0). The neglect of 1 /Nc corrections in the 1 /Nc expansion makes sense for the comparison with the scattering picture in the large Nc limit, described in the following section. One can see that the Hamiltonian (2) reproduces the characteristic Nc scaling for the excitation energy of baryons which is N0 [2]. The nucleon case In large Nc the color part of the wave function is antisymmetric so that the orbital-spin-flavor part must be symmetric to satisfy the Pauli principle. A quanta of orbital excitation requires the orbital part to be mixed symmetric, the lowest state having the partition [Nc — 1,1]. We have the following [Nc — 1,1] spin-flavor (SF) states which form a symmetric state with the orbital l = 3 state of partition [Nc — 1,1] 1. [Nc_l)l]SF=[N^l)N^l]sx[^,^]F, Nc > 3 with S = 1/2 and J = 5/2,7/2 2. [Nc-1,1]SF=p^,^]sx Nc > 3 with S = 3/2 and J = 3/2,5/2,7/2,9/2. They give rise to matrices of a given J either 2 x 2 or 1 x 1 depending on the multiplicity of J. States of symmetry [Nc — 1,1]SF with S = 5/2, like for A (see below), which together with l = 3 could give rise to J = 11/2, are not allowed for N, by inner products of the permutation group [22]. Therefore the experimentally observed resonance N(2600)Ii ! /2 should belong to the N = 5 band (l = 5). For Nc =3 the above states correspond to the 2 8 and 4 8 multiplets of SU(2) x SU(3) respectively. The A case In this case the Pauli principle allows the following states 1. [Nc-UIsf^Y^IS^^^IF- Nc > 3 with S = 1/2 and J = 5/2,7/2, 2. [NC-1,1]SF=[^,^]Sx Nc > 5 with S = 3/2 and J = 3/2,5/2,7/2,9/2, 3. [Nc-1,1]SF=p^,^]sx Nc > 7 with S = 5/2 and J = 1 /2,3/2,5/2,7/2,9/2,11 /2. As above, they indicate the size of a matrix of fixed J for the Hamiltonian (2). For example, the matrix of A5/2 is 3x3, because all three states can have J = 5/2. For Nc = 3 the first state belongs to the 210 multiplet. The other two types of states do not appear in the real world with Nc = 3. Note that both for N j and Aj states the size of a given matrix equals the multiplicity of the corresponding state indicated in Table 1 of Ref. [21] for l = 3. The Hamiltonian (2) is diagonalized in the bases defined above. Let us denote the eigenvalues either by m{N' or m^' with i = 1, 2 or 3, depending on how many eigenvalues are at a fixed J. The Hamiltonian has analytical solutions, all eigenvalues being linear functions in the coefficients c, c2 and c3. It is remarkable that the 18 available eigenstates with I = 3 fall into three degenerate multiplets, like for I = 1. If the degenerate masses are denoted by m2, m3 and m4 we have (3) (4) (5) m2 = mi11 = mN1 N 3/2 = mi1 a3/2 = mN1 N 5/2 = mi1 i5/ 2 = mi1 i7/2 m3 (2) = mi a3/2 (2) = mN N 5/2 (2) = mi a5/2 = mN1 N 7/2 (21 = mi i7/2 = mi1 i9/2 m4 (3) = mi a5/2 (2) = mN N 7/2 (3) = mi a7/2 = mN1 N 9/2 (21 = mi i9/2 = mi1 i 11/2 where m2 m3 m4 = ci Nc CiNc - -2C2--C3, 1 15 2C2 + Ï6C3' 3 5 ciNc + -c2 - — c3. (6) (7) (8) 2 " 16 The notation m^ is used to distinguish this eigenvalue from m2 of Ref. [19]. In the following subsection we shall see that the scattering picture gives an identical pattern of degeneracy in the quantum numbers, but the resonance mass is not quantitatively defined. Therefore only a qualitative compatibility can be established. 2.2 The meson-nucleon scattering picture Here we are concerned with nonstrange baryons, as above, and look for a degeneracy pattern in the resonance picture. The starting point in this analysis are the linear relations of the S matrices S^l'RRqj and SLRj of n and n scattering off a ground state baryon in terms of K-amplitudes. They are given by the following equations [19,21] S?L'RR'IJ = LHV/(2R + D(2R' + 1)(2K + 1) { R/ lT/ J } { R [ 1 / (9) Slrj = h § klmLRJ)Sk> (10) and where sKL'L and sK are the reduced amplitudes. The notation is as follows. For n scattering R and R' are the spin of the incoming and outgoing baryons respectively (R =1/2 for N and R = 3/2 for A), L and L' are the partial wave angular momentum of the incident and final n respectively (the orbital angular momentum L of n remains unchanged), I and J represent the total isospin and total angular momentum associated to a given resonance and K is the magnitude of the grand spin K = I + J. The 6j coefficients imply four triangle rules §(LRJ), §(R11), §(L1 K) and §(IJK). These equations were first derived in the context of the chiral soliton model [23,24] where the mean-field breaks the rotational and isospin symmetries, so that J and I are not conserved but the grand spin K is conserved and excitations can be labelled by K. These relations are exact in large Nc QCD and are independent of any model assumption. The meaning of Eq. (9) is that there are more amplitudes S^l'rr 'j than there are s£LL' amplitudes. The reason is that the IJ as well as the RR' dependence is contained only in the geometrical factor containing the two 6j coefficients. Then, for example, in the nN scattering, in order for a resonance to occur in one channel there must be a resonance in at least one of the contributing amplitudes s£LL'. But as skll' contributes in more than one channel, all these channels resonate at the same energy and this implies degeneracy in the excited spectrum. From the chiral soliton model there is no reason to suspect degeneracy between different K sectors. From the meson-baryon scattering relations (9) and (10) three sets of degenerate states have been found for I =1 orbital excitations [19]. There is a clear correspondence between these sets and the three towers of states [19,20] of the excited quark picture provided by the symmetric core + excited quark scheme [9]. They correspond to K = 0,1 and 2 in the resonance picture. But the resonance picture also provides a K = 3 due to the amplitude s j22. As this is different from the other skl'l , in Ref. [19] it was interpreted as belonging to the N = 3 band. Here we extend the work of Ref. [19,21] to 1 = 3 excited states which belong to the N = 3 band. The partial wave amplitudes of interest and their expansion in terms of K-amplitudes from Eqs. (9) and (10) can be found in Tables I-III of Ref. [16]. They correspond to L = L' = 2, L = L' = 4 and L = L' = 6 respectively. From those tables one can infer the following degenerate towers of states with their contributing amplitudes Al/2 > n3/2> A3/2, n5/2 a5/2 > n7/2 > associated to K = 2,3,4,5 and 6 respectively. We can compare the towers (11)-(15) with the quark-shell model results of (3)-(5). The first observation is that the agreement of (11) (K = 2) with (3), of (12) (K = 3) with (4) and of (13) (K = 4) with (5) is perfect regarding the quantum numbers. Second, we note that the resonance picture can have poles with K = 5,6 which infer the towers (14) and (15). They have no counterpart in the quark-shell picture for I = 3. But there is no problem because the poles with K = 5,6 can belong to a higher band, namely N = 5 (£ = 5) without spoiling the compatibility. Comparing these results with those of Ref. [21] one can conclude that one can associate a common K = 2 to 1 = 1 and 1 = 3. For this value of K the triangular rule 6(K£1) proposed in Ref [21] is satisfied. The quark-shell picture brings however more information than the resonance picture due to the fact that it implies an A3/2, N5/2, A5/2, A7/2, (s222, .s2), (11) 1 A5/2 , N7/2 , a7/2 , A9/2, (s322, 1s344), (12) A7/2, N9/2, A9/2, a11/2 > (s444, .sn), (13) A7/2, N9/2, A9/2, a11/2 > (s544, is566), (14) A9/2, a11 /2 > (s666, ,sn6) (15) energy dependence via the l dependence which measures the orbital excitation. Note that m^ is different from m2 of l = 1 [19,20]. Because in the resonance picture they stem from the same amplitude s^, one expects that this amplitude possesses two poles at two distinct energies, in order to have compatibility. Thus the number of poles of the reduced amplitudes s£LL remains an open question. We anticipate that a similar situation will appear for every value of K associated to two distinct values of l, satisfying the 6(Kl1) rule, for example, for K = 4 which is common to l = 3 and l = 5. 3 Conclusions We have compared two alternative pictures for baryon resonances consistent with large the Nc QCD limit and found that the two pictures are compatible for l = 3 excited states, as it was the case for l = 1. The quark-shell picture is practical and successful in describing known resonances and in predicting other members of the excited octets and decuplets. But the extended symmetry SU(2Nf) x O(3) where O(3), which is essential to include orbital excitations, does not have a direct link to large Nc. On the other hand the scattering picture is close to experimental analysis but it is not clear where the pole positions should lie. It is however very encouraging that the two pictures give sets of degenerate states with identical quantum numbers when one works at order O (NC). It is a qualitative proof that the spin-flavor picture is valid and useful for baryon phenomenology. References 1. G. 't Hooft, Nucl. Phys. 72 (1974) 461. 2. E. Witten, Nucl. Phys. B160 (1979) 57. 3. J. L. Gervais and B. Sakita, Phys. Rev. Lett. 52 (1984) 87; Phys. Rev. D30 (1984) 1795. 4. R. Dashen and A. V. Manohar, Phys. Lett. B315 (1993) 425; ibid B315 (1993) 438. 5. R. F. Dashen, E. Jenkins and A. V. Manohar, Phys. Rev. D51 (1995) 3697. 6. E. Jenkins, Ann. Rev. Nucl. Part. Sci. 48 (1998) 81; AIP Conference Proceedings, Vol. 623 (2002) 36, arXiv:hep-ph/0111338; PoS E FT09 (2009) 044 [arXiv:0905.1061 [hep-ph]]. 7. D. Pirjol and T. M. Yan, Phys. Rev. D 57 (1998) 1449. 8. J. L. Goity, Phys. Lett. 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Suppl. 174 (2007) 155. 19. T. D. Cohen and R. F. Lebed, Phys. Rev. Lett. 91, 012001 (2003); Phys. Rev. D67 (2003) 096008. 20. D. Pirjol and C. Schat, Phys. Rev. D67 (2003) 096009. 21. T. D. Cohen and R. F. Lebed, Phys. Rev. D68 (2003) 056003. 22. F. Stancu, "Group theory in subnuclear physics," Oxford Stud. Nucl. Phys. 19 (1996) 1. 23. A. Hayashi, G. Eckart, G. Holzwart and H. Walliser, Phys. Lett. 147B (1984) 5. 24. M. P. Mattis and M. E. Peskin, Phys. Rev. D32 (1985) 58; M. P. Mattis, Phys. Rev. Lett. 56 (1986) 1103; Phys. Rev. D39 (1989) 994; Phys. Rev. Lett. 63 (1989) 1455; M. P. Mattis and M. Mukerjee, Phys. Rev. Lett. 61 (1988) 1344. Bled Workshops in Physics Vol. 12, No. 1 p. 64 News from Belle M. Bracko* University of Maribor, Smetanova ulica 17, SI-2000 Maribor, Slovenia and J. Stefan Institute, Jamova cesta 39, SI-1000 Ljubljana, Slovenia Abstract. This paper reports on some of the latest spectroscopic measurements performed with the experimental data collected by the Belle spectrometer, which has been operating at the KEKB asymmetric-energy e+ e- collider in the KEK laboratory in Tsukuba, Japan. 1 Introduction The Belle detector [1] at the asymmetric-energy e+ e- collider KEKB [2] has accumulated about 1 ab-1 of data by the end of its operation in June 2010. The KEKB collider, called a B-factory, most of the time operated near the Y(4S) resonance, while at the end of its operation it was running mainly at the Y(5S) resonance. Large amount of collected experimental data and excellent detector performance enabled many interesting spectroscopic results, including discoveries of new hadronic states and studies of their properties. This report covers most recent and interesting spectroscopic measurements—performed with either charmonium(-like) and bottomonium(-like) states. 2 Charmonium and Charmonium-like States 2.1 and nc(2S) in B meson decays There has been a renewed interest in charmonium spectroscopy since 2002. The attention to this field was drawn by the discovery of the two missing cc states below the open-charm threshold, nc(2S) and hc(1P) [3,4] with JPC=0 + and 1 + , respectively. Still, many questions about the lightest charmonium states have been unanswered. For example, the width of the nc(1S) has been determined with large discrepancies between experiments with different production mechanisms: in J/^ and ^(2S) radiative decays rnc~15 MeV, while in B meson decays oryy—>nc processes, F^c ~30 MeV [5]. In a recent Belle analysis [6] a data sample of 535 million of BB pairs is used for the study of B+—>K+r|c(—>KsK±7tT) decays1. The mass and the width of the nc were determined by a 2-dimensional fit of the invariant mass * Representing the Belle Collaboration. 1 In this review, the inclusion of charge-conjugated states is always implied. Tnc [MeV] Production Mechanism Measured by 35.1±3.1i];j? B decays Belle [6] 30.5±1.0±0.9 Tj/^yric BESIII [7] 28.1±3.2±2.2 yy^ric Belle [8] 31.7±1.2±0.8 yy^ric BaBar [9] 36.3i|;£±4.4 B decays BaBar [10] Table 1. Recent measurements of the r|c width. Minv(KSKn) vs. the angle between Ks and K+ from B+ in the nc centre-of-mass system. Since nc is a pseudoscalar meson, the angular distribution should be flat, but significant P- and D-wave components from non-resonant charmless B background decays are also observed. By including the above angle into the fit, the interference with the background seems to be correctly taken into account, and as a result the measured nc width, listed in Table 1, is found to be consistent with other recent measurement. The nc mass is determined to be (2985.4±1.5-0 0) MeV. The same study [6] is performed also for the nc(2S) meson. For this first radially excited 0~+ cc state the width measurement is important, because the potential model predictions are less reliable due to the vicinity of the D°D° threshold. The analysis shows, that here the interference with the non-resonant background is even larger as in the case of the nc. The measured width is rnc(0S) = (6.6-8 . 1-0 . 6) MeV for the fit with interference and (41.1±12.0-f049) MeV, when the interference is not taken into account, i.e. for the fit of the invariant mass only. The factor 5 smaller width of the nc(2S) when compared to the nc can be explained only by the wave function differences, since both states decay hadron-ically via two gluons. With the new measurement, the error on the world average of the nc(2S) width is decreased for almost a factor of 2. 2.2 The X(3872) news The story about new charmonium-like states (so called "XYZ" states) began in 2003, when Belle reported on B+ —> K+J/^n+n- analysis, where a new state decaying to J/^n+n- was discovered [11]. The new state, called X(3872), was soon confirmed and also intensively studied by the CDF, D0 and BABAR collaborations [12-20]. So far it has been established that this narrow state (r = (3.0-] ^ ± 0.9) MeV) has a mass of (3872.2 ± 0.8) MeV, which is very close to the D°D*° threshold [5]. The intensive studies of several X(3872) production and decay modes suggest two possible Jpc assignments, 1++ and 2~+, and establish the X(3872) as a candidate for a loosely bound D°D*° molecular state. However, results provided substantial evidence that the X(3872) state must contain a significant cc component as well. Recently, Belle performed a study of B —> (ccy)K using the final data sample with 772 million of BB pairs collected at the T(4S) resonance [21], Pure D°D*° Experiment [Reference] Measured X(3872) mass [MeV] CDF [24] 3871.61±0.16±0.19 BaBar (B+) [25] 3871.4±0.6±0.1 BaBar (B°) [25] 3868.7±1.5±0.4 D0 [12] 3871.8±3.1±3.0 Belle [23] 3871.84±0.27±0.19 LHCb [26] 3871.96±0.46±0.10 Updated World Average 3871.67±0.17 Table 2. Measurements of the X(3872) mass. First error is due to limited statistics, while the second corresponds to systematic uncertainties. molecular model [22] predicts B(X(3872) ^'y) to be less than B(X(3872) J/^Y). Results by the BABAR collaboration [20] show that B(X(3872) ^'y) is almost three times that of B(X(3872) —> J/^y), which is inconsistent with the pure molecular model, and can be interpreted as a large cc — D°D*° admixture. We observe X(3872) —> J/^y together with an evidence for xc2 —> J/^Y in B± —> J/^decays, while in our search for X(3872) —> ^'y no significant signal is found. We also observe B —> Xci K decays in both, charged as well as neutral B decays. The obtained results suggest that the cc-D°D*° admixture in X(3872) may not be as large as discussed above. New results for the X(3872) J/^n+n- decay modes in B+—>K+X(3872) and B°—>K° (—m+n-)X(3872) decays are obtained with the complete Belle data set of 772 million BB pairs collected at the T(4S) resonance [23]. The results for the X(3872) mass and width are obtained by a 3-dimensional fit to distributions of the three variables: beam-constrained-mass Mbc= ^(E™^)2 — (p™8)2 (with the beam energy Ebmsm and the B-meson momentum pBms both measured in the centre-of-mass system), the invariant mass Minv(J/^n+n-) and the energy difference AE=EBms—Ebmfm (where EBms is the B-meson energy in the centre-of-mass system). As a first step, the fit is performed for the reference channel ^'—> J/^n+n-, and the resolution parameters are then fixed for the fit of the X(3872). The mass, determined by the fit, is listed in Table 2 in comparison to other precise measurements. Including the new Belle result, the updated world-average mass of the X(3872) is mx=(3871.67±0.17) MeV. If the X(3872) is an S-wave D*°D° molecular state, the binding energy Eb would be given by the mass difference m(X)—m(D*°)-—m(D°). With the current value of m(D°)+m(D*°)=(3871.79 ± 0.30) MeV [5], a binding energy of Eb=(—0.12±0.35) MeV can be calculated, which is surprisingly small and would indicate a very large radius of the molecular state. The best upper limit for the X(3872) width was 2.3 MeV (with 90% C.L.), obtained by previous Belle measurement [11]. The 3-dimensional fits are more sensitive to the natural width, which is smaller than the detector resolution (a ~4 MeV). Due to the fit sensitivity and the calibration performed on the reference channel ^'—> n , the updated upper limit for the X(3872) width is about 1/2 of the previous value: r(X(3872)) < 1.2 MeV at 90% C.L. Previous studies performed by several experiments suggested two possible JPC assignments for the X(3872), 1++ and 2 +. In the recent Belle analysis [21], the X(3872) quantum numbers were also studied with the full available data sample collected at the Y(4S) resonance. Although at the current level of statistical sensitivity it is not possible to distinguish completely between the two possible quantum number assignments, the study shows that quantum numbers JPC=1++ seem to be slightly preferable for the X(3872) state. 3 Bottomonium and Bottomonium-like States An interesting question is whether in the bb systems there exist analogous "XYZ" states, predicted by many of the models proposed to explain the charmonium-like exotic states. Also, even for regular bottomonium states there are a lot of unanswered questions. Some of the answers are expected to be given by analyses of the Belle data sample of 121 fb-1, collected at the energy of the Y(5S) resonance. The Belle collaboration used a data sample at the CM energy around the Y(5S) mass 10.89 GeV, and found large signals for decays into n+ n-Y(1S), n+n-Y(2S) and n+n-Y(3S) final states [33]. If these transitions are only from the Y(5S) resonance, then the corresponding partial widths are between 0.5 and 0.9 MeV. These values are more than two orders of magnitude larger than the corresponding partial widths for Y(4S), Y(3S) and Y(2S) decays to n+n-Y(1S). Recent CLEO-c results for the process e+e- —> hc(1P)n+ n- showed that its rate is comparable to the process e+e- —> J/ij)7t+7t- at a/s = 4170 MeV and found an indication of even higher transition rate at the Y(4260) energy [34]. Analogously, these results imply that the hb (mP) production might be enhanced in the region of the Yb and motivate a search for the hb(mP) in the Y(5S) data. hb(1P) and hb (2P) states are observed in the missing mass spectrum of n+n- pairs for the Y(5S) decays, with significances of 5.5ct and 11 .2ct, respectively [35]. This is the first observation of the hb (1P) and hb (2P) spin-singlet bottomonium states in the reaction e+ e- —> hb(mP)n+n- at the Y(5S) energy. Comparable rates of hb(1P) and hb(2P) production indicate a possible exotic process that violates heavy quark spin-flip and this motivates a further study of the resonant structure in Y(5S) —> hb(mP)n+n- and Y(5S) —> Y(nS)n+ n-decays [36]. Due to the limited statistics, only the study of M(hb (mP)n) distribution is possible for hb(mP)n+n-, while in the case of Y(nS)n+n- decay modes the Dalitz plot analysis can be performed. As a result, two charged bottomonium-like resonances, Zb( 10610) and Zb (10650), are observed with signals in five different decay channels, Y(nS)n± (n = 1,2,3) and hb(mP)n± (m = 1,2). The averaged values for the mass and widths of the two states are calculated to be: M(Zb(10610)) = (10608.4 ± 2.0) MeV, r(Zb(10610)) = (15.6 ± 2.5) MeV and M(zb (10650)) = (10653.2 ± 1.5) MeV, r(Zb (10650)) = (14.4 ± 3.2) MeV. The measured masses are only a few MeV above the thresholds for the open beauty channels B*B (10604.6 MeV) and B*B* (10650.2 MeV), which could indicate a molecular nature of the two observed states. Angular analysis of charged pion distributions favors the JP = 1+ spin-parity assignment for both Zb( 10610) and Zb (10650). 4 Summary and Conclusions The Belle experiment at the KEKB collider provides an excellent environment for charm and charmonium spectroscopy. As a result, many new particles have already been discovered during the Belle operation, and some of them are mentioned in this report. Some recent Belle results also indicate that analogs to exotic charmonium-like states can be found in bb systems. As the operation of the experiment has just finished in June 2010, more interesting results on charmoni-um(-like) and bottomonium(-like) spectroscopy can still be expected from Belle in the near future. References 1. Belle Collaboration, Nucl. Instrum. Methods A 479,117 (2002). 2. S. Kurokawa and E. Kikutani, Nucl. Instrum. Methods A 499,1 (2003), and other papers included in this Volume. 3. Belle Collaboration, Phys. Rev. Lett. 89,102001 (2002). 4. Cleo Collaboration, Phys. Rev. Lett. 95,102003 (2005). 5. K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). 6. Belle Collab., arXiv:1105.0978v2 [hep-ex], to appear in Phys. Lett. B. 7. BESIII Collab., preliminary results, presented at XIV International Conference on Hadron Spectroscopy (Hadron2011), Munich, Germany. 8. Belle Collaboration, Eur. Phys. J. C 53,1 (2008). 9. BaBar Collaboration, Phys. Rev. D 81, 052010 (2010). 10. BaBar Collaboration, Phys. Rev. D 78, 012006 (2008). 11. Belle Collaboration, Phys. Rev. Lett. 91, 262001 (2003). 12. CDF Collaboration, Phys. Rev. Lett. 93,072001 (2004); D0 Collaboration, Phys. Rev. Lett. 93,162002 (2004); BaBar Collaboration, Phys. Rev. D 71, 071103 (2005). 13. Belle Collaboration, arXiv:hep-ex/0505037, arXiv:hep-ex/0505038; submitted to the Lepton-Photon 2005 Conference. 14. Belle Collaboration, Phys. Rev. Lett. 97,162002 (2006). 15. BaBar Collaboration, Phys. Rev. D 74, 071101 (2006). 16. Belle Collaboration, arXiv:0809.1224v1 [hep-ex]; contributed to the ICHEP 2008 Conference. 17. Belle Collaboration, arXiv:0810.0358v2 [hep-ex]; contributed to the ICHEP 2008 Conference. 18. CDF Collaboration, Phys. Rev. Lett. 98,132002 (2007). 19. BaBar Collaboration, Phys. Rev. D 77, 011102 (2008). 20. BaBar Collaboration, Phys. Rev. Lett. 102,132001 (2009). 21. Belle Collaboration, Phys. Rev. Lett. 107, 091803 (2011). 22. E. S. Swanson, Phys. Rep. 429, 243 (2006). 23. Belle Collaboration, Phys. Rev. D 84, 052004(R) (2011). 24. CDF Collaboration, Phys. Rev. Lett. 103, 152001 (2009). 25. BaBar Collaboration, Phys. Rev. D 77,111101(R) (2008). 26. LHCb Collab., Proc. XIX International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2011), LHCb-CONF-2011-021. 27. Belle Collaboration, Phys. Rev. Lett. 100,142001 (2008). 28. BaBar Collaboration, Phys. Rev. D 79,112001 (2009). 29. Belle Collaboration, Phys. Rev. D 80, 031104 (2009). 30. Belle Collaboration, Phys. Rev. D 78, 072004 (2008). 31. BaBar Collaboration, Phys. Rev. D 74, 091103 (2006). 32. BES Collaboration, Phys. Rev. Lett. 100,102003 (2008). 33. Belle Collaboration, Phys. Rev. Lett. 100,112001 (2008); Phys. Rev. D 82, 091106 (2010). 34. CLEO-c Collaboration, Phys. Rev. Lett. 107, 041803 (2011). 35. Belle Collab., arXiv:1103.3419 [hep-ex], submitted to Phys. Rev. Lett. 36. Belle Collaboration, arXiv:1105.4583 [hep-ex]. Bled Workshops in Physics Vol. 12, No. 1 p. 70 Electroproduction of mesons in a chiral quark model B. Golli Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia and Jozef Stefan Institute, 1000 Ljubljana, Slovenia Motivation This work is a continuation of a joint project on the description of baryon resonances performed by the Coimbra group (Manuel Fiolhais, Luis Alvarez Ruso, Pedro Alberto) and the Ljubljana group (Simon Sirca and B. G.) The pion- and photon-induced meson production on nucleons are important tools to study the hadron dynamics in the first and second resonance region. One of the main challenges is to understand the interplay of quark and meson degrees of freedom. While several models of nucleon excited states spanning from the non-relativistic models based solely the quark degrees of freedom to models involving only mesonic degrees of freedom are able to successfully describe the pion elastic and non-elastic scattering in the resonance region, electro-production of mesons represents a much more severe test which may be able to disentangle the properties originating in the (valence, constituent) quark degrees of freedom from those of the meson cloud. One of the most widely recognized example is electro-excitation of the A(1232) resonance where the pion cloud contributes ~ 45 % to the magnetic dipole amplitude, and strongly dominates the electric quadrupole amplitude. Similarly, the behaviour of the pion electro-excitation amplitude in the Roper region can be explained by assuming a relatively strong contribution of the pion cloud. However, in the second resonance region, such a conclusion is less transparent because of the presence of other channels. To avoid ambiguities, we need to develop a method in which the strong and the EM processes are treated in a unified approach without too many adjustable parameters. The method In order to study the interplay of quark and meson degrees of freedom, we have developed a method that incorporates the nucleon and its excited states calculated in different chiral quark models into a coupled channel approach involving different meson-baryon channels as well as the photon-nucleon channel. The conceptual foundations of our approach in chiral quark models date back to the paper [1], in which we demonstrated the above mentioned importance of the pion cloud in electro-production of pions in the region of the A(1232). In [2] we have generalized our approach used in our previous studies of the resonances (see e.g. [1], [3], [4] and [5]). The generalized method incorporates excited baryons represented as quasi-bound quark-model states into a coupled channel framework using the K-matrix formalism. It can be applied to meson scattering as well as to electro and weak-production of mesons. Our method assumes a class of chiral quark models in which mesons couple linearly to the quark. In such a case it is possible to write down an exact expression for the K matrix (and, consequently, for the T matrix) in terms of the principal-value states corresponding to the meson-baryon channels possessing the proper asymptotic behaviour. The construction respects the symmetry of the K matrix and hence ensures the unitarity of the S matrix. The strong and the weak points The main advantages of our method can be summarized in • Baryons are treated as composite particles from the very beginning; the strong and electro-weak form-factors are derived from baryon internal structure and not inserted a posteriori; as a consequence the method introduces a much smaller number of free parameters. • The physical resonances appear as linear superpositions of bare resonances. • The bare quark-meson and quark-photon vertices are modified through meson loops as well as through mixing of resonances and coupling to the background. • The meson cloud around baryons is included in a consistent way also in the asymptotic states. • The method yields a symmetric K matrix and hence respects the unitarity of the S matrix. The present limitations of the method are primarily the absence of meson-meson interaction and the nucleon-meson four-point interaction which can be introduced only in an approximate way. This is a consequence of our assumption about the meson-quark interaction discussed above. Our method is therefore primarily intended to describe the processes in the region of resonances rather than in the energy region close to the threshold where other methods are anyway superior. The Roper resonance In [2] and [6] we have considered the scattering and the pion electro-production in the region of the N (1440) and of its I = | partner, the A(1600). As the underlying quark model we have taken the Cloudy bag Model, primarily because of its simplicity. A good agreement with the observed scattering amplitude and the Mi_ electro-production amplitude is found provided the nA and the oN channels of comparable strength are included in the multichannel calculation. The results strongly support the hypothesis that the pion cloud plays an important role in the case of the electro-excitation of N* (1440) resonance, especially in the region of low Q2 (long-range effects). In this region the quark contribution is small and positive, while the pion contribution and the vertex corrections due to meson loops are large and negative. At intermediate Q2, these two effects are responsible for the zero crossing of the amplitude. At higher Q2 (short-range physics) the quark core takes over, rendering the amplitude positive. The S11 resonances In recent years there have been substantial efforts to understand the peculiar nature of the lightest of the S11 resonances, the N(1535), due to its position just above the nN threshold and the large branching ratio to the nN channel. The extension of the approach to the low-lying negative-parity resonances requires the inclusion of new channels involving the s- and d-wave pions, the n and the p mesons, and the KA channel. In [7-9] we have used an SU(3) extension of the Cloudy Bag Model taking fn and fK from the meson sector, while for the other model parameters we have used the same values as in the case of the positive-parity resonances, adding only the mixing parameter between the two bare-quark states corresponding to N(1535) and N(1650), and their bare masses. We have obtained a good overall agreement with the available experimental results for the partial widths of the N(1535) and the N(1650) resonances as well as for the pion, n-meson and kaon electroproduction amplitudes. In particular, the excellent agreement with the data for n production strongly supports our conjecture about the dominance of the genuine three-quark configuration in the N*(1535) state. While the cross-section for pion-induced production of K+ appears to be over-estimated in our model, the photo-production amplitude is smaller than predicted by phenomenological analyses. This discrepancy remains an open question and represents a challenge for further investigation. References 1. M. Fiolhais, B. Golli, S. Sirca, Phys. Lett. B 373, 229 (1996) 2. B. Golli and S. Sirca, Eur. Phys. J. A 38, (2008) 271. 3. P. Alberto, M. Fiolhais, B. Golli, and J. Marques, Phys. Lett. B 523, 273 (2001). 4. B. Golli, S. Sirca, L. Amoreira, M. Fiolhais Phys.Lett. B553 (2003) 51-60 5. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, and S. Sirca, Eur. Phys. J. A 26, 99 (2005). 6. B. Golli, S. Sirca, and M. Fiolhais, Eur. Phys. J. A 42,185 (2009) 7. B. Golli, S. Sirca, Eur. Phys. J. A 47 (2011) 61. 8. B. Golli, talk given at the Sixth International Workshop on Pion-Nucleon Partial-Wave Analysis and the Interpretation of Baryon Resonances, 2327 May, 2011, Washington, DC, U.S.A., http: //gwdac . phys . gwu . edu/pwa2 011/Thursday/b_golli . pdf 9. Simon Sirca, Bojan Golli, Manuel Fiolhais and Pedro Alberto, in Proceedings of the XIV International Conference on Hadron Spectroscopy, 13-17 June 2011 Munich, Germany, edited by S. Paul, N. Brambilla, and B. Grube, http://arxiv.org/abs/110 9.0163. Bled Workshops in Physics Vol. 12, No. 1 p. 73 Scattering phase shift and resonance properties* S. Prelovseka,b, C. B. Langc and D. Mohlerd a J. Stefan Institute, Jamova 39,1000 Ljubljana, Slovenia b Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia c Institut für Physik, FB Theoretische Physik, Universitat Graz, A-8010 Graz Austria d TRIUMF, 4004 Wesbrook Mall Vancouver, BC V6T 2A3, Canada Abstract. We describe the method for extracting the elastic scattering phase shift from a lattice simulation at an introductory level, for non-lattice practitioners. We consider the scattering in a resonant channel, where the resulting phase shift §(s) allows the lattice determination of the mass and the width of the resonance from a Breit-Wigner type fit. We present the method for the example of P-wave nn scattering in the p meson channel. 1 Introduction The determination of the strong decay width of a hadronic resonance in lattice QCD is a much more demanding task than the determination of its approximate mass. The only available method (that was applied up to now) was proposed by Luscher [1] and is rather indirect. It applies for the case when the resonance appears in the elastic scattering of two hadrons Hi H2 —> R —> Hi H2. • First, the energy spectrum En of the system of two interacting hadrons H1 H2 enclosed in a few-fermi box has to be determined. The system is illustrated in Fig. 1. The spectrum in a finite box En is discrete and few (one or two) lowest energy levels have to be determined by lattice simulation. • The shift of the energy En with respect to the non-interacting energy EHi (pi) + Eh2 (P2) (EHi(pi) = \/mi + Pi2) gives info on the interaction between Hi and H2. Luscher derived a rigorous relation between the energy shift En — EH1 — EH2 and the elastic phase shift 6(s) for H1H2 scattering in continuum [1]. The measured energies En can be used to extract the phase shift §(s) evaluated at s = En — P2, where En is the energy of the system and P its total momentum. In order to extract 6(s) at several different values of s, the simulations are done for several choices of total momenta P of the H1H2 system, which leads to different values of s = E^n — P2. • The resulting dependence of §(s) as a function of s can be used to extract the mass mR and the width rR of the resonance R, which appears in the elastic channel H1 H2 —> R —> H1 H2. For this purpose, the 6(s) can be fitted with a Breit-Wigner form or some other phenomenologically inspired form, which depend on mR and rR. * Talk delivered by S. Prelovsek Pi ;p2 L >> few fm JP2 L= few fm Fig. 1. The energy of two hadrons in a box of size L. On the left, L ^ fm and E(L) ~ Ehi (p1) + EH0 (p2). On the right, L ~ few fm and energy gets shifted due to their interaction, i.e. E(L) ~ Ehi (p1) + Eho(p2) + AE(L). The described method, needed for the determination of the resonance width rR, is rather challenging. It requires very accurate determination of a few lowest energy levels of the system Hi H0, since the resulting phase shift depends ultimately on the energy shift. Among all the meson resonances, this method has been up to now rigorously applied only to p resonance. Although Liischer proposed the method already in late 80's [1], the first lattice attempt to employ it to hadronic resonances had to wait until 2007 [2]. Since then, several studies of p have been carried out [3,4], with the most up to date ones [5-7]. This talk briefly describes the method to extract 6(s), mR and rR on an example of nn scattering in the p channel. It is based on a recent simulation [6], which is the statistically most accurate determination of any strong meson width on one lattice ensemble. The purpose of this talk is to highlight the main physical reasoning, which lies behind the lattice extraction of 6(s), mR and rR, omitting most of technical details. The sections follow the order of steps required, which are listed as items in the introduction. Section II describes the determination of spectrum En of the coupled system H1H0 <-> R. The Section III described why En allow one to extract the elastic phase shift 6(s). The extraction of the resonance parameters mR and rR from the phase shift 6(s) is done in Section IV. We end with conclusions. 2 Spectrum of two hadrons in a finite box The p meson is a resonance in nn scattering in P-wave, and has quantum numbers IG(JPC) = 1 + (1 ). The total momentum P of the coupled nn — p system can have values -^-d , d e Z3 due to the periodic boundary condition in the spatial direction, and we use the following three choices P= (0,0,0) , ^(0,0,1) , ^(1,1,0) and permutations . (1) This enables us to obtain several values of s = E^ — P0 for the system, thereby allowing the determination of 6(s) for these values of s without changing the spatial volume. Our simulation is performed on an ensemble of 280 [8] gauge configurations with dynamical u/d quarks, where the valence and dynamical quarks employ improved Wilson-Clover action. The corresponding pion mass is mna = 0.1673 ± 0.0016 or mn = 266 ± 4 MeV. The lattice spacing is a = 0.1239 ± 0.0013 fm and we employ a rather small volume N^ x NT = 163 x 32, which allows us to use the costly full distillation method [9] for evaluating the quark contractions. On the lattice, the discrete energies of the system En can be extracted after computing the dependence of the correlation matrix Ctj (tf,tt) on Euclidean time tf - tt Ctj(tf,ti) = (0Oi(tf) 0|(ti)|0> = ^(OiJnXnlOf> e-En(tf-ti) . (2) n The analytical expression on the right is obtained by inserting the complete set Hn ln> (nl of physical states n with given quantum numbers. The interpolators Ot have the quantum numbers of the system in question. In our case the interpolators have quantum numbers JPC = 1 and |I, I3> = |1,0> and total three-momentum P. They have to couple well to the nn state and the quark-antiquark resonance p. For each choice of P (1), we use 16 interpolators, listed in detail in Eq. (21) of [6]. We employ fifteen interpolators of quark-antiquark type Oiqq(t) = eiP" fa-FiU (t,x) + dfid (t,x)] , (3) x where Ft denotes different color-spin-space structures with the same resulting quantum number JPC = 1 and |I,I3> = 11, 0>. We use also one n(pi)n(p2) interpolator, where each pion is projected to a definite momentum On"(t) = ^[7T+(p1)7T-(p2)-7T-(p1)7T+(p2)] , Pl + p2 = P , n±(pi)= X etPiX qY5T±q (t, x) (4) x In practice, the nn interpolator is the most important among our 16 interpolators, since it couples to the scattering state much better than the quark-antiquark interpolators. Let us note that all other lattice studies aimed at rp used at most one quark-antiquark and one nn interpolator, which may not always allow for reliable extraction of the first excited energy level E2. Given the 16 interpolators, we compute the 16x16 correlation matrix Ctj (tf, tt) for all initial and final time-slices tt,tf = 1, ..,NT = 32. The needed Wick contractions that enter the correlation matrix with our q q and nn interpolators are depicted in Fig. 2. The contributions (a,c,e) in Fig. 2 cannot be evaluated solely from the quark propagator from one point (tt, xt) to all other points of the lattice (such a propagator allowed most of the spectroscopy studies in the past). The contributions (a,c,e) require the propagators from all and to all points on the lattice, which is too costly to evaluate in practice. We use the recently proposed distillation method for this purpose [9], which enables the exact computation of the required contractions. (a) (c) Fig. 2. Contractions for I = 1 correlators with q q (3) and nn (4) interpolators. p level n En a sa2 6 ^f (0,0,0) tM0,0,0) 1 2 0.5107(40) 0.9002(101) 0.2608(41) 0.8103(182) 130.56(1.37) 146.03 (6.58) [*] TMO,O,1) tMO,O,1) 1 2 0.5517(26) 0.6845(49) 0.1579(29) 0.3260(69) 3.06 (0.06) 156.41(1.56) ^(1,1,0) ^(1,1,0) 1 2 0.6933(33) 0.7868(116) 0.1926(49) 0.3375(191) 6.87(0.38) 164.25(3.53) Table 1. The results for two lowest levels n = 1,2 of the coupled nn — p system with three choices of total momentum P on our lattice with mna = 0.1673 ± 0.0016, L = 16a and the lattice spacing a = 0.1239 ± 0.0013 fm. The energy levels En are obtained by multiplying En a with a-1 ~ 1.6 GeV. The invariant mass squared of the system is s = E^ — P2, but the dimensionless value in the table s a2 is obtained using the discretized version of this relation [6]. We average the resulting correlators (i) over all initial time slices tt at fixed time separation tf — tt, (ii) over all directions of momenta P (1) and (iii) over all directions of the p meson polarization. The time dependence tf — tt of the correlators Ctj (tf, tt) (2) contains the information on the energies of the system En, and several methods for extracting En from Ctj are available. We extract two lowest energy levels En=1,2 of the system from the 16 x 16 correlation matrix Ctj (tf, tt) using the so called variational method [10], which is the most established among the available methods. Table 1 displays the extracted lowest two energies En=1,2 of the coupled nn — p system for our three choices of total momenta P (1). The spectrum En in Table 1 for our finite box is the main result of this section. Each energy level corresponds to a different value of s = E^ — P2, as calculated from En and P in the Table 1. In fact, the table lists values of s obtained from the discrete lattice version of the dispersion relation, which takes into account part of the corrections to s = E^ — P2 due to finite lattice spacing [6]. 3 Extraction of the phase shifts from energy levels Let us consider the case when the resonance R can strongly decay only to two spinless hadrons Hi and H2, so one has elastic scattering of Hi and H2. We point out that the non-elastic case, when a resonance can decay strongly to several final states (i.e. H1H2 and H' H2), is much more challenging for a lattice study. Suppose one encloses two hadrons H1 (p1) H2(p2) with three-momenta p1 and p2 into a large box of size L ^ fm and measures their energy. In a large box, they hardly interact and their energy is equal to sum of individual energies Enon-int = EHi(pi)+EH2(p2) with EH(p) = \Jm^ + P2- Now, let's force H^pO and H2(p2) to interact by decreasing the size of the box to L of a few fm. The energy of the system E(L) = EH1 (p1) + EH2(p2) + AE(L) is shifted with respect to Enon-int: it will increase (AE(L) > 0) if the interaction is repulsive and decrease (AE(L) < 0) if the interaction is attractive. This simple physical reasoning indicates that the energy shift AE(L) gives info on the interaction. Fig. 3. The scattering of two interacting particles as series of the interaction vertex M(6L) and the scattering of non-interacting particles F at finite L [11]. In fact, the energy shift AE(L) and the energy itself E(L) do not only give us "some" info on the interaction. According to the seminal analytic work of Liischer [1], E(L) or AE(L) rigorously tells us the value of the elastic scattering phase shift of H1 H2 scattering at L —> oo, i.e. 6(L = oo): Luscher method : E(L)—> 6(s,L = oo) s = E(L)2 — P2 (5) The derivation and the resulting formulae between E(L) and 6 are lengthy and rather complicated, but let us briefly explain at least why E(L) contains info on 6(L = oo). A nice and clear quantum-filed theory derivation is given in [11] and the main message is illustrated in Fig. 3. The scattering of two interacting spin-less hadrons Hi H2 at finite L (for degenerate case mHi = mH2 = m) is represented in QFT by series of: • scattering of two non-interacting hadrons at finite L, represented by F. The expression F contains sums over the loop momenta k, which are allowed in a finite box L with periodic boundary conditions in space. Here f (k0, k) stands for dependence of the vertices on the left and right on k0 and k. • the interaction vertex M with four hadron legs. This vertex depends on the elastic phase shift 6^ (at infinite volume) for the case of elastic scattering in the l-th partial wave. The physical scattering requires resummation of the bubbles in Fig. 3, with non-interacting parts F and the interacting parts M, giving AF^j^A'. The positions of the poles of the sum AF j J^A' obviously depend on M and therefore on 6i. The positions of the poles dictate the possible energy levels of the system En(L), so the energy levels En(L) depend on M and therefore on 6^. The purpose of the above illustration was just to indicate why En(L) depend on 6^. In the case of nn with JP = 1the relevant wave has l = 1 and we denote the corresponding phase by 6 = 6i. The complete analytic relations between En(L) and 6(s) needed for our case of the nn scattering with JPC = 1 and I = 1 are provided in [6] (for every |P| a different form of relation applies). These allow to extract 6 for each of our six energy levels in Table 1 and the resulting phase shifts are given in the same Table. The presented Lüscher formalism applies only for the case of elastic scattering. The nn state is the only scattering state in this channel for energies when 4n state cannot be created, i.e., when s = E^ < (4mn)2. For our mna = 0.1673 this is valid for all six levels, with the exception of the level Ei at P = 0, which is above 4n inelastic threshold. As the Lüscher analysis is not valid above the inelastic threshold, we omit this level from further analysis. The resulting scattering phase shifts for five values of s are shown in Fig. 4. This is the main result of the lattice study; the resonance properties will be obtained by fitting 6(s) in the next section. Note that the resulting phases are determined with a relatively good precision, which is better than in other available lattice studies of p at comparable u/d quark masses. The good precision can be traced back to various advanced techniques we used: the distillation method for evaluating contractions, usage of a large interpolator basis and average over all initial time slices, directions of momenta P and polarizations of p. 4 Extracting resonance mass and width from the phase shift The phase shift 6(s) in Fig. 4, obtained directly from the lattice study, can be used to extract the properties of the resonance, in our case the p. The phase shift has a typical resonance shape: it passes from 6 ~ 0° to 6 ~ 180°: the point where it crosses 90° gives the position of the resonance (s = mp), while the steepness of the rise gives its width rp. In particular, 6 is related to resonance parameters by expressing the scattering amplitude a^ in terms of 6 on one hand, and with Breit-Wigner form in the vicinity of the resonance on the other hand — a/s F(s) e2i6(s) _ 1 Q1 = -2 , • rr( ^ = -T-- • (6) s — mp + rys I (s) 2i Relation (6) can be conveniently re-written as v^Hs) cot6(s) =m2 -s . (7) co 180 I-r 16014012010080604020- 0.1 0.15 0.2 0.25 s 0.3 0.35 0.4 Fig. 4. The nn phase shift 6 (s) (in degrees) for five different values of dimensionless sa2 = (Ena)2 — (Pa)2, extracted from our lattice study [6]. The s is obtained by multiplying sa2 with (a-1)2 ~ (1.6 GeV)2. lattice (this work [6]) exp [PDG] ttw ~ 266 MeV TTVp 792 ± 12 MeV 775 MeV 9 P7T7T 5.13 ±0.20 5.97 Table 2. Our lattice results for the resonance parameters [6], compared to the experimental values. The decay width significantly depends on the phase space and therefore on mn, so the decay width extracted at mn ~ 266 MeV could not be directly compared to the measured width. So, it is customary to extract the p —> nn coupling gpnn instead of the width, where the width p*3 g2 r(s) rP = rK) (8) depends on the phase space for a P-wave decay and the coupling gpnn. The coupling is expected to be only mildly dependent on mn, which was explicitly confirmed in the lattice studies [5,7] and analytic study [12]. In (8), p* denotes the pion momentum in the center-of-momentum frame and we extract it from s using a discretized version of relation a/s = 2y/m2 + p*2 [6]. Inserting P(s) (8) into (7), one obtains an expression for 6(s) in terms of two unknown parameters: mp and gpnn. We fit these two parameters using five values of 6(s) given in Fig. 4 and Table 1, and we get the values of resonance parameters in Table 2 with small statistical errors. The resulting p-meson mass in Table 2 is slightly higher than in experiment, as expected due to mn = 266 MeV > mnxp. The coupling gpnn is rather close to the value gpnt derived from the experimental width r^xp. Fig. 5. The crosses are the nn phase shift 6(s) (in degrees) for five different values of di-mensionless sa2 = (Ena)2 — (Pa)2, extracted from our lattice study [6]. The line is the Breit-Wigner fit (7,8) for the resulting mp and gpnn in Table 2. The physical value of s is obtained by multiplying sa2 with (a-1 )2 ~ (1.6 GeV)2. 5 Comparison to other lattice and analytical studies The comparison of our results for mp and rp to two recent lattice studies [5,7] is compiled in Fig. 8 of [7]. Our result has the smallest error on a given ensemble, demonstrating that accurate lattice determination mR and rR for (some) resonances is possible now. The other two lattice studies are done for two [7] and four [5] pion masses and explicitly demonstrate mild dependence of gpnn on mn. The discussion concerning the (dis)agreement of the three lattice studies is given in [7] and will be extended in [13]. The comparison of our 6(s) to the prediction of the lowest non-trivial order of unitarized Chiral Perturbation Theory [14] is given by the solid line in 6, which has been recalculated for our mn = 266 MeV in [15]. The lowest1 order prediction does not depend on unknown LECs and agrees reasonably well with our lattice result, given by the bullets. 6 Conclusions We highlighted the main physical reasoning, which lies behind the lattice extraction of elastic phase shifts 6(s) and the resonance parameters mR and rR. The purpose was to present the general principle of the method and omit the technical details. The method was presented on the example of nn —> p —> nn scattering. This example demonstrates that a proper first-principle treatment of some hadronic resonances on the lattice is now possible. 1 One cannot make a fair comparison between out lattice result and the next-to-lowest order prediction, since it depends on a number of LECs, and some of them have been fixed using mp from another lattice study, which gets a significantly higher mp . iso 100 120 140 80 20 40 so 0 150 - -1 PI; _----1 PI; --Ur ------Ur - ............Ur 011 nn+ and p —> pn0 fluctuation in agreement with the observed value in the (e + p —> e+ forward neutron+X) experiment. 1 Introduction First we review some evidence for the role of the pion cloud in explaining nucleon observables. As examples of low-energy processes, we quote the magnetic polar-izabilities [1] and electroexcitation of the nucleon [2-4]. The pion cloud acts as a coil and gives a diamagnetic contribution while the virtual excitation of the N-like quark core into the A-like quark core acts as a paramagnet. The magnetic polar-izability of the nucleon results from an approximate cancellation between these two contributions. Without the pion cloud, the paramagnetic contribution would dominate and give much too large magnetic polarizability. In the electroexcita-tion of the nucleon into A and into the Roper resonance, the linear a-model with quarks and the cloudy bag model help us understand why (40 - 50)% of the dominant M1 ampliutude and 100% of the E2 amplitude is due to pion cloud. The question arises, whether the same amplitude of the pion cloud (or equiv-alently, the same probability of pion fluctuation) can explain also observables measured at higher energies where the stucture functions of quarks play a role and pion is seen through its contribution to the corresponding quark and antiquark structure function. 2 Pion cloud in quarks can explain nucleon observables The notion of the constituent quark applies generally to the massive quark dressed by gluons, the constituent of the nucleon. This non-relativistic model with three * Talk delivered by M. Rosina massive constituent quarks works well for the hadronic masses and the magnetic moments. It breaks down if the spin properties of the baryons are considered. The improved version, the chiral constituent quark model is surprisingly suc-cesful in explaining the spin properties of nucleons and hyperons. In the simplest form applied to the nucleon the chiral constituent quark is composed of a massive quark accompanied by a quark-antiquark pair coupled to the spin-parity quantum numbers of the pion Jn = 0-. In the following we write the pion symbol as a shortcut to the quark-antiquark pair coupled to the pion quantum numbers. This simple model has been first applied by Eichten et al. [5] to explain the flavor asymmetry of the sea quarks and further elaborated by Baumgartner et al. [6] and Pirner [7] in the interpretation of the spin properties of the nucleon. It is related to the three-flavour extension proposed by Cheng and Li [8]. Explicitly written, the chiral constituent up-quark (u) structure is |u> = \J( 1 - |a) |u) - V^|d7t+) + |U7T°), (1) and of the down quark (d) Id) = - | a) Id) + v^Iutt-) - ^||d7t°). (2) The basis of pure flavour quarks is denoted by boldface u and d. At Q2 « 0 gluons do not appear as an explicit degree of freedom and the nucleon is composed of quarks and quark-antiquark pairs. Thus in the lowest order the Fock state of the constituent quark has the form (1 and 2), where in the second and third term the quark-antiquark pair is coupled to the Jn = 0- quantum nummbers of the pion. This simple structure of the chiral constituent quark (1) has two attractive features. Firstly, as we will show, the chiral constituent quark reproduces the experimental results of the deep inelastic scattering and axial-vector beta decays of the neutron quantitatively; secondly, this model complies with our picture of the origin of the quark mass by the chiral symmetry breaking mechanism of Nambu and Jona-Lasino [9]. Dressing the light quark by gluons is inevitably accompanied by creation of the Goldstone boson, the pion. The Goldstone pion is an inherent part of the constituent quark. The parameter a of (1, 2) is usually determined from the value of the axial vector coupling constant gA = 1.269 ± 0.003 [11] yielding a = 0.239 ± 0.002. The parameter a measures the probability of the constituent quark to be in the state accompanied with a charged pion. Furthermore, with the probability a/2 the constituent quark is in a state component with the neutral pion. Thus the total probability of finding a pion in the constituent quark amounts thus to slightly more than one third. The large probability of the pion in the constituent quark is best manifested in the measurements of the quark polarization in the deep inelastic scattering. Not only that one third of the constituent quark with the pion does not contribute to the spin polarization, but even more, with the oppositely oriented quarks reduces the total quark polarization to one third of what would be without the pions. The loss of the angular momentum because of the oppositely oriented quark is compensated by the orbital angular momentum of the pion in the p-state. The comparison of the experimental results of the deep inelastic scattering with the prediction of the chiral constituent-quark model is given in [10]. It is also worthwhile to mention that the valence-quark distribution does not peak at Bjorken x = 0.3 but it is softer and peaks at x = 0.2 corresponding to five and not three constitunts of the proton even before gluons can get excited. Eichten et al. ([5]) ascribe these quark-antiquark pairs to an asymmetric sea. We consider also other observables which depend strongly on the pions in the nucleon: the Gottfried sum rule IG (with corrections discussed in [10]), the integrals of the spin structure functions of proton Ip and deuteron Id and the quark spin polarization AI [12,13]. They have larger error bars than gA, but they agree reasonably well eith the model (Table 1.). The new experimental value for AI supports even more our assumption that the main contribution to the spin reduction comes from the pion fluctuation. observable model value 9a = : 1.269 ± 0.003 #C1 - a) input ig = 0.216 ± 0.033 - 2a) = 0.174 ± 0.002 iP = 0.120 ± 0.017 fsV - 2a) = 0.145 ± 0.002 Id = 0.043 ± 0.006 36 1 1 - 3a) = 0.039 ± 0.001 AI = : 0.330 ± 0.064 (1 - 3a) = 0.283 ± 0.006 Table 1. The probability a = 0.239 ± 0.002 is used to calculate different observables 3 The proton contains a neutron plus pion component Let us consider the matrix element (nn+|p). Inserting for constituent quarks our chiral quarks it is evident that the (nn+| has an overlapp with a Fock component of the proton. The result of the explicit calculation is 3 |(n7t+|p)|2 = |(d7t+|u)|2 = (1 - - a) a = 0.15. (3) The result (3) means that the constituent u quark has a component of the d quark and a pion. Although the proton has two u quarks there is no factor 2 in the amplitude, due to the flavor-spin-color structure of the nucleon. The flavor-spin wavefunction of the proton has a mixed symmetry combined into a symmetric flavor-spin function: P = 1 2 3 1 2 [Ï + V 2 S 1 3 X f 1 3 3 2 2 s (4) x f A similar expression stays for the neutron. Since the combined wavefunction is symmetric under all permutations it is enough to look at the contribution of the particles 1 and 2. In the first term of the proton wavefunction the particles 1 and 2 are symmetric and can both be u quarks and contribute constructively to the matrix element with a factor of two. In the second term the interference is destructive and the contribution cancels. Thus only the first term contributes to the matrix element. Since both in proton and in neutron the first term appears with a factor \f\Jl, the factor two is canceled out. This qualitative explanation can be verified by writing down the three-quark wavefunctions explicitly. This can be seen even easier in the isospin formalism. In the act of producing a positive pion, the corresponding u quark loses one unit of charge, it becomes a d quark. This can be described with the operator Y.i t-(i) = T_ where T_ = Tx — iTy. We conveniently took the sum over all three quarks since the third quark, d, contributes zero anyway. The expectation value is < TM — 1 |T_|TM >= ^JT[T + 1) - M(M — 1) which for proton (T = 1 /2, M = 1/2) gives in fact the factor 1. It is instructive to compare with A+ (T = 3/2, M = 1 /2) in the process ep—> eAX where one gets the factor 2, pointing out that the two u quarks are always symmetric and interfere constructively. Of course, for the squared amplitude, we get the additional factor a since only the n+-dressed component of the u-quark contributes, and the factor (1 — |a) for the naked component of the final d-quark. 4 Experimental test of the pion fluctuation The pion fluctuation of nucleon is well known in the classical nuclear physics as anomalously large pion-nucleon coupling constant g2/4n = 13.6. Many of the nucleon properties are ascribed to the pion cloud of the nucleon [14]. Hovewer, there is no direct way of determinig experimentally the probability of finding a pion fluctuation in the proton. The best way is to calculate the pion flow by using the pion-nucleon coupling constant and the form factor assuming that the pion is emitted by a proton [15], [16] = "(sirtjilGWl1. (5) The pion flow is related to the measured cross section by dCTy*p^nX = fn+/p(xL,t) • X (6) where the (y*n+ —> X) DIS cross section is assumed to be 2/3 of the (y*p —> X) DIS cross section in the cited analysis, with corrections due to absorption [10]. Obviously the pion is not emitted by a proton but by a quark. But as we showed above the state of the pion is dictated by the proton wave function and the pion form factor simulated well the assumption that the emission is from the proton. In the series of experiments [17]- [18], [19] measuring the spectrum of the forward neutrons in the reaction (e+p—>e+forward n+X) has been shown that the high energy end of the neutron spectrum is consistent with the assumption that the deep inelastic scattering takes place on the pion. Thus we are justified to say that the forward neutron is the signature of the reaction taking place on the pion and that the total probability of finding a pion in ep—mn+ fluctuation can be obtained by integrating over the variables of the pion flow. The analysis depends to some extent on the estimation of pion flux fn+ /p. The analysis has been elaborated in [10] and the quoted results are (nn+|p)2 = 0.165 ± 0.01 and 0.175 ± 0.01, respectively, for the two form factors best fitting to the experiment in [15] and [16]. 5 Conclusion The pion fluctuation p—rn+n+ and p—>p+n0 is an artifact of the quark-antiquark pairs of the constituent quarks. The impressive agreement between the measured and the calculated ratios between the probability of the pion fluctuation and the probability of finding a quark-antiquark pair of the constituent quark is a strong support of the constituent quark model. In this section we stress the difference between the notion of the quarkantiquark pairs coupled to the pion quantum numbers being part of the constituent quarks and the pions of the proton. While the quark-antiquark pairs are implied by the experimental values of gA, the integrated spin structure functions and the violation of the Gottfried summ rule, the fluctuating pions are identified by the characteristic energy and pT distribution of the neutron spectra in the ep —> nn+ reaction. Eichten et al. [5] have named the quark-antiquark pairs of the constituent quark the asymmetric quark sea. This name emphasizes hopefully sufficiently the difference of their origin as compared to the normal quark sea. For the value a = (dn+|u)2 = 0.24 each quark contains 0.36 quark-antiquark pairs. Summing up the quark-antiquark pairs one obtains about one quark-antiquark pair per nucleon. Using this value of a gives (nn+|p)2 = 0.15. This number corresponds well with the experimental value of (nn+|p)2 = 0.165 ± 0.01 or 0.175 ± 0.01. It follows that in « 0.26 cases the proton is a neutron+ or a proton+n0. This means that about one quater of the nucleon's quark-antiquark pairs show up as the pion fluctuation. Acknowledgments We wish to thank A. Bunyatyan for the discussions of the analysis on the forward neutron spectra and K. Rith for pointing us out the new results on spin-polarization data in DIS. References 1. B. Golli and R. Sraka, Phys. Lett. B 312 (1993) 24-29. 2. M. Fiolhais, B. Golli and S. Sirca, Phys. Lett. B 373 (1996) 229-234. 3. B. Golli and S. Sirca, Eur. Phys. J. A 38 (2008) 271-286. 4. B. Golli, S. Sirca and M. Fiolhais, Pion electro-production in the Roper region in chiral quark models, Eur. Phys. J. A 42 (2009) 185-193. 5. E.J.Eichten, I.Hinchliffe and C.Quigg, Phys. Rev. D 45 (1992) 2269. 6. S. Baumgaertner, H. J. Pirner, K. Koenigsmann, B. Povh, Z. Phys. A 353 (1996) 397. 7. H. J. Pirner, Prog. Part. Nucl. Phys. 36 (1996) 19-28. 8. T. P. Cheng and Ling-Fong Li, Phys. Rev. Lett. 74 (1995) 2872-2875. 9. S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 694. 10. A. Bunyatyan and B. Povh, Eur. Phys. J. A 27 2006 359-364. 11. K. Nakamura et al. (Particle Data Group), J. Phys. G 37 (2010) 075021. 12. K. Rith, Prog. Part. Nucl. Phys. 49 (2002) 245. 13. A. Airapetian et al., Phys. Rev. D 75 (2007), 012007. 14. A. W. Thomas, Prog. Theor. Phys. 168 (2007) 614 15. B. Kopeliovich, B. Povh and I. Potashnikova, Z. Phys. C 73 (1996) 125, [hep-ph/9601291]. 16. H. Holtmann et al., Phys. Lett. B 338 (1994) 363. 17. C. Adloff et al. [H1 Collaboration], Eur. Phys. J. C 6 (1999) 587, [hep-ex/9811013]. 18. A. Aktas et al. [H1 Collaboration], Eur. Phys. J. C 41 (2005) 273, [hep-ex/0501074]. 19. S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 610 (2005) 199, [hep-ex/0404002], and references therein. Bled Workshops in Physics Vol. 12, No. 1 p. 88 Recent experimental results from MAMI (Mainz), ELSA (Bonn), and JLab* S. Sircaa,b a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia b Jozef Stefan Institute, Ljubljana, Slovenia Abstract. In the past year a large set of new data on photo- and electro-production of mesons on nucleons and light nuclei has emerged, both near threshold and throughout the nucleon resonance region. Some of the most recent results from the three leading experimental facilities, MAMI (Mainz, Germany), ELSA (Bonn, Germany), and Jefferson Lab (Newport News, USA) relevant to this workshop are presented. 1 n and n photo-production on protons In conjunction with the development of polarized target techniques and polarime-try capabilities, production of single mesons by real photon beams has recently become the richest source of information on nucleon dynamics from the meson production threshold and throughout the nucleon resonance region. The basic quantities that can be measured when polarization is exploited, are: when the photon beam and the target are both unpolarized, and when the beam is unpolarized and the target is polarized along the y-direction, respectively; I, H , P , G , with linearly polarized beam and no target polarization, and with target polarized along x-, y- and z-directions, respectively; F , E , when the beam is circularly polarized and the target polarization is along the x- and z-directions, respectively. (The z-axis points along the beam; the z- and x-axes span the meson production plane; the y-axis is perpendicular to it.) In addition to resolving the spin (helicity) structures, so-called complete experiments can be performed for which measurements of the same observables have to be performed in different reaction channels, so that isospin decomposition can be done as well. Such measurements are also underway. * Talk delivered by S. Sirca ct,T , With the Crystal Ball and TAPS detectors at ELSA, preliminary results for I (beam-helicity asymmetry) in the Yp —> pn0 and Yp —> pn processes have been obtained at EY = 1050 MeV as a function of the pion emission angle 0 [1,2]. Almost the complete angular range has been covered at this energy. Apart from a few modest unresolved deviations between the ELSA and older GRAAL data sets [3] at extreme backward angles, all data are in excellent agreement with the MAID and SAID analyses, as well as the Bonn-Gatchina Partial-Wave Analysis (PWA). Moreover, there is a new precise unpolarized data set for n photoproduction from MAMI [4] which provides precise cross sections up to 400 MeV above threshold. Very sturdy results have been obtained on the angular expansion coefficients Ai, A2, and A3, which will be of great help in improving various PWA. On the other hand, measurements of I in single-pion photo-production have also been performed at Jefferson Lab within the CLAS Collaboration in the g8b group of experiments, and here the agreement with respect to the theory (in particular MAID) is not as good. Largest deviations are observed at forward angles where I is typically underestimated by theory. Because both final channels on the proton target (pn0 and nn+) have been measured, different sensitivities to N* and A resonances could be probed. This is a very comprehensive and large data set encompassing all angles and photon energies from 1000 to 2000 MeV. At ELSA, I has also been determined at 0 = 110° as a function of EY ranging from about 700 to 1200 MeV, indicating that the Pii(1440), D13(1520) and F15 (1680) resonances are all needed (at least within the MAID model) to reproduce the energy dependence of I. Most recently, similar-quality results have been obtained at ELSA for the double-polarization asymmetry G in Yp —;1 pn (energy dependence at 0 = 110° ), as well as for G in the Yp ~> pn process. In the pn channel, at EY = 950 and 1050 MeV, the preliminary data on G appears to be in rough agreement with the Bonn-Gatchina PWA and MAID, but is underestimated by SAID at 1050 MeV. 2 Electro-excitation of nucleon resonances The focus of investigations of nucleon resonance excitations has recently shifted away from the Delta region to the first and second resonance regions. The bulk of the new data comes from the CLAS Collaboration and EBAC (Excited Baryons Analysis Center) at Jefferson Lab. The most spectacular advances have been made regarding the Roper resonance N*(1440). The transverse and scalar helicity couplings extracted from a wealth of previous single-pion production data [5,6] and the most recent two-pion data set [7] are in excellent mutual agreement. In the framework of the EBAC analysis, this allows for a model-independent determination of the N* electromagnetic couplings for Q2 up to « is now clear that the transverse helicity amplitude Ai /2 crosses zero in the vicinity of Q2 « 0.5 GeV2 and that the structure of the Roper can evidently be explained in terms of a quark core as a first radial excitation of three dressed quarks, plus external meson-baryon dressing. (The CLAS12 project will test these findings to much higher Q2 « 12 GeV2.) There are also new data on helicity amplitudes for the electro-excitation of the N*(1535) resonance on the proton, extracted from both the nn+ and the pn channel, both of which, again, are in good agreement between each other in the transverse case (while there are no pn data in the scalar case). Note that this is the first extraction ever of S1/2(Q2) up to Q2 « 4GeV2. With the present data on A1 /2 extending to relatively high values of Q2, it is now possible to investigate (or rather, speculate) whether transitions to the regime of perturbative QCD occur. The main motivation behind these scaling studies is to observe the transition to photon interactions with the dressed quarks. The Q2 dependence of the product Q3A1 /2(Q2) has been studied as function of Q2 in the P11 channel (Roper), the S11 channel (N*(1535)) and the D13 channel (N*(1520)). Apparently Q3A1/2(Q2) flattens out at Q2 as low as « 3GeV2, persisting to Q2 « 4 GeV2 where the data ceases. But although this plateau is appealing, extensions to higher Q2 are needed to confirm it. 3 n, nn, and n photo-production on deuterons Most interesting experiments have been performed on the deuteron, in particular single-n0 and single-n photo-production. The reaction mechanisms for n° photoproduction are For Ey > 500 MeV, the coherent contribution is negligible. Practically all measurements focus on that region, where the sum of the exclusive processes on the proton and the neutron should almost exactly add up to the quasi-free inclusive result. This process has been previously measured at MAMI/A2 [8] and LNS Sendai [9], but has now been superseded by a much lovelier data set [10]. Single-meson production on deuterons has important ramifications regarding the inclusion of D13, F15, and D15 resonances in unitary-isobar models and partial-wave analyses, as the proton and neutron channels exhibit distinct sensitivities to these ingredients. Two-pion production on the proton and the deuteron (allowing for the extraction of the corresponding neutron channel contribution) is relevant in the very same sense [11]. New preliminary data on yp —> pn°n° and yn —> nn°n° from the CB/TAPS @ MAMI Collaboration has become available, indicating that the electro-magnetic excitation of the F15 is relatively stronger on the proton, while the excitation of the D15 is stronger on the neutron. Helicity asymmetries for these two processes have also been measured. But of the recent data sets, one of the most exciting and puzzling is that on quasi-free n photo-production on the deuteron, acquired by the CB/TAPS Collaboration at ELSA. The differential cross-section for this process exhibits a narrow structure at W « 1.65 GeV, regardless of the n0 + p(n) ; quasi — free on p , n0 + n(p) ; quasi — free on n , n0 + d ; coherent. Y + d -» n + n(p) , emission angle (specifically, it has been most often shown at -0.9 < cos 0 < -0.5, -0.3 < cos 0 < 0.1, and 0.1 < cos 0 < 0.5). The structure has been observed previously at LNS Sendai, by the GRAAL Collaboration [12], but now high precision data by the CBELSA/TAPS Collaboration has become available [13], clearly identifying the structure at W = 1660 MeV , r =(25 ± 12) MeV . Because the excess cross-section appears in the rescaled neutron cross-section as compared to the free-proton and quasi-free proton cases, this structure has become known as the "neutron anomaly". The reasons for the anomaly remain unknown, although several explanations have been offered. It may be an interference effect of the S11(1650) and P11(1710) resonances; it may be caused by a non-strange penta-quark; but it could also be generated by a KI threshold enhancement of the neutron cross-section as a consequence of the pion loops. Note that while the np cross-section is « 80 % S-wave, the nature of the nn is not so well known. At ELSA, quasi-free n photo-production has also been measured on 3 He nuclei. The idea behind replacing the deuteron by 3He is that these nuclei have different nucleon momentum distributions (in the deuteron case, it peaks at « 40 MeV, while it is maximal at « 70 MeV in 3He). This should generate different proton/neutron cross-section contributions in the processes Y +3 He —> n + p(pn) , Y +3 He —> n + n(pp) . Apparently the broad structure at W « 1.65 GeV persists, with r = (45 ± 11) MeV, which is comparable to the experimental resolution. References 1. D. Elsner et al. (CB/TAPS @ ELSA Collaboration), Eur. Phys. J. A 39 (2009) 373. 2. A. Thiel et al. (CB/TAPS @ ELSA Collaboration), submitted for publication. 3. O. Bartalini et al. (GRAAL Collaboration), Eur. Phys. J. A 26 (2005) 399. 4. E. F. McNicoll et al. (CB @ MAMI Collaboration), Phys. Rev. C 82 (2010) 035208. 5. M. Dugger et al. (CLAS Collaboration), Phys. Rev. C 79 (2009) 065206. 6. I. Aznauryan, V. Burkert et al., Phys. Rev. C 80 (2009) 055203. 7. V. Mokeev, AIP Conf. Proc. 1374 (2011) 349. 8. B. Krusche et al. (A2 Collaboration), Eur. Phys. J. A 6 (1999) 309. 9. H. Shimizu, NNR Workshop 2009, Edinburgh, June 8-10, 2009 (unpublished). 10. M. Dieterle et al. (A2 Collaboration), contribution to the Hadron 2011 Conference, Munich, June 13-17, 2011, arXiv:1108.6241 [nucl-ex]. 11. H. Arenhovel, A. Fix, L. Tiator, Eur. Phys. J. A 25 (2005) 115. 12. A. Kuznetsov et al. (GRAAL Collaboration), Phys. Lett. B 647 (2007) 23. 13. I. Jaegle et al. (CBELSA/TAPS Collaboration), Eur. Phys. J. A 47 (2011) 89. Blejske Delavnice Iz Fizike, Letnik 12, št. 1, ISSN 1580-4992 Bled Workshops in Physics, Vol. 12, No. 1 Zbornik delavnice 'Razumevanje hadronskih spektrov', Bled, 3. - 10. julij 2011 Proceedings of the Mini-Workshop 'Understanding hadronic spectra', Bled, July 3-10, 2011 Uredili in oblikovali Bojan Golli, Mitja Rosina, Simon Sirca Publikacijo sofinancira Javna agencija za knjigo Republike Slovenije Tehnični urednik Tadeja Sekoranja ZaloZilo: DMFA - zaloZnistvo, Jadranska 19,1000 Ljubljana, Slovenija Natisnila tiskarna Bori v nakladi 120 izvodov Publikacija DMFA stevilka 1853 Brezplacni izvod za udelezence delavnice