BLEJSKE DELAVNICE IZ FIZIKE Bled Workshops in Physics ISSN 1580-4992 L ETNIK 5, S T. 1 VOL. 5, NO. 1 Proceedings of the Mini-Workshop Quark Dynamics Bled, Slovenia, July 12-19,2004 Edited by Bojan Golli Mitja Rosina Simon Sirca University of Ljubljana and JoZef Stefan Institute DMFA - ZALOZNIŠTVO Ljubljana, november 2004 The Mini-Workshop Quark Dynamics was organized by Joief Stefan Institute, Ljubljana Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana and sponsored by Ministry of Education, Science and Sport of Slovenia 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 Jose Amoreira, Covilha, amoreira@dfisica.ubi.pt Enrique Ruiz Arriola, Granada, earriola@ugr.es Wojciech Broniowski, Krakow, b4bronio@cyf-kr.edu.pl Luciano Canton, Padova, luciano.canton@pd.infn.it Manuel Fiolhais, Coimbra, tmanuel@teor.fis.uc.pt Harald Fritzsch, München, fritzsch@mppmu.mpg.de Leonid Glozman, Graz, leonid.glozman@uni-graz.at Dubravko Klabuöar, Zagreb, klabucar@phy.hr William Klink, Iowa, william-klink@uiowa.edu Thomas Melde, Graz, thomas.melde@uni-graz.at Willibald Plessas, Graz, plessas@bkfug.kfunigraz.ac.at Dan-Olof Riska, Helsinki, riska@pcu.helsinki.fi Bojan Golli, Ljubljana, bojan.golli@ijs.si Damijan Janc, Ljubljana, damijan.janc@ijs.si Mitja Rosina, Ljubljana, mitja.rosina@ijs.si Simon ¡Sirca, Ljubljana, simon.sirca@fmf.uni-lj.si Tomo Zivko, Ljubljana, tomi.zivko@ijs.si Electronic edition http://www-f1.ijs.si/BledPub/ Contents Preface............................................................. V Polyakov Loop at Finite Temperature in Chiral Quark Models E. Megias, E. Ruiz Arriola, and L.L. Salcedo............................. 1 Application of chiral quark models to high-energy processes W. Broniowski and E. Ruiz Arriola..................................... 7 On the Meson-Few-Body Problem From a Few-Nucleon Perspective L. Canton and L. G. Levchuk.......................................... 11 Quark matter and quark stars M. Fiolhais, L. P. Linares, M. Malheiro, A. Taurines....................... 18 A Time Dependence of the Scale of QCD H. Fritzsch.......................................................... 22 Critical Review of Pentaquarks L. Ya. Glozman...................................................... 27 Excited Hadrons on the Lattice L. Ya. Glozman (for BGR collaboration)................................. 28 A heuristic derivation of an effective QCD coupling dominated by gluon condensates D. Kekez and D. Klabucar............................................. 29 Vertex Interactions and Applications W H. Klink......................................................... 36 Strong Decays of Baryons T. Melde, L. Canton, W. Plessas, and R. F. Wagenbrunn ................... 43 Relativistic Treatment of Baryon Reactions W. Plessas .......................................................... 49 The Double-Charm Hyperons and Their Interactions D.-O. Riska ......................................................... 58 Calculation of electroproduction amplitudes in the K-matrix formalism B. Colli, P. Alberto, L. Amoreira, M. Fiolhais and S. Sirca.................. 62 Molecular binding of Tcc = DD* tetraquark D. Janc,M. Rosina ................................................... 70 New ideas about production and detection of cc-tetraquarks M. Rosina, D. Janc................................................... 74 Recent results on A resonance production at MIT-Bates, MAMI, and JLab (Hall A) S. Sirca............................................................. 77 Search for Pentaquarks at HERA-B T. Zivko............................................................ 83 Preface This summer's Mini-Workshop on Quark Dynamics has been another in the traditional series of meetings held at Bled, in the inspiring atmosphere of Villa Ple-melj. The Workshop, virtually free of the time constraints imposed on speakers at large-scale venues, has retained the spirit of "friendly confrontation" among physicists working on closely related problems in hadronic physics. With respect to the previous Workshops, the emphasis has shifted from the structure of hadrons to the dynamics of their production and detection, and a colorful set of topics has been covered. The relativistic approach has been advanced one step further, using the spectator approximation and the point form. Yet, a few "naughty" electro-magnetic and mesonic decays of baryons remain to be open problems. To see, or not to see a pentaquark was an unbalanced issue with prevailing arguments against the sightings. On the other hand, the tetraquark proponents were optimistic about the conclusion that the DD* state is probably bound. The Roper resonance has been observed in lattice QCD. Production of pions was shown to be a three-body problem sensitive to spin-orbit and tensor forces. Does the strong coupling change over time? Maybe a laser can tell. High energies, high temperatures, high densities, the chiral phase transition, and quark stars still excite our phantasy. Can effective interactions be parameterized directly by Feynman graphs? What is the role of the gluon condensate? These Proceedings represent a succinct record of the broad range of issues discussed at the Workshop. Ljubljana, November 2004 M. Rosina B. Colli S. Sirca Workshops organized at Bled > What Comes beyond the Standard Model (June 29-July 9,1998) Bled Workshops in Physics 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) Bled Workshops in Physics 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) Bled Workshops in Physics 2 (2001) No. 1 > What Comes beyond the Standard Model (July 17-27,2001) Bled Workshops in Physics 2 (2001) No. 2 > Studies ofElementarySteps ofRadical Reactions in Atmospheric Chemistry > Quarks and Hadrons (July 6-13,2002) Bled Workshops in Physics 3 (2002) No. 3 > What Comes beyond the Standard Model (July 15-25,2002) Bled Workshops in Physics 3 (2002) No. 4 > Effective Quark-Quark Interaction (July 7-14,2003) Bled Workshops in Physics 4 (2003) No. 1 > What Comes beyond the Standard Model (July 17-27,2003) Bled Workshops in Physics 4 (2003) No. 2-3 > Quark Dynamics (July 12-19,2004) Bled Workshops in Physics 5 (2004) No. 1 > What Comes beyond the Standard Model (July 19-29,2004) 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 Bled Workshops in Physics 3 (2002) No. 1-2 Polyakov Loop at Finite Temperature in Chiral Quark Models* E. Megías, E. Ruiz Arrióla, and L.L. Salcedo Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain Abstract. At finite temperature, chiral quark models do not incorporate large gauge invariance which implies genuinely non-perturbative finite temperature gluonic degrees of freedom. Motivated by this observation, we describe how the coupling of the Polyakov loop as an independent degree of freedom to quarks not only accounts for large gauge invariance, but also allows to establish in a dynamical way the interaction between composite hadronic states such as Goldstone bosons to finite temperature non-perturbative gluons in a medium which can undergo a confinement-deconfinement phase transition. 1 Large Gauge Transformations One feature of gauge theories like QCD at finite temperatures in the imaginary time formulation [1-3] is the non-perturbative manifestation of the non Abelian gauge symmetry. In the Polyakov gauge, where 94A4 = 0 and A4 is a diagonal and traceless Nc x Nc matrix, and Nc is the number of colors, there is still some freedom in choosing the gluon field. Let us consider for instance the periodic gauge transformation [4,5] g(x4) = ei2nX4A/p , (1) where A is a color traceless diagonal matrix of integers. We call it a large gauge transformation (LGT) since it cannot be considered to be close to the identity1. The gauge transformation on the A4 component of the gluon field is A4^A4 + yA. (2) Thus, invariance under the LGT, Eq. (1), implies a constant shift in the A4 gluon amplitudes, meaning that A4 is not uniquely defined by the Polyakov gauge condition. These ambiguities on the choice of the gauge field within a given gauge fixing are usually called Gribov copies. The requirement of gauge invariance actually implies identifying all amplitudes differing by a multiple of 2n/|, which means periodicity in the diagonal amplitudes of A4 of period 2n/|. Perturbation theory, which corresponds to expanding in powers of small A4 fields manifestly breaks gauge invariance at finite temperature, because a Taylor expansion on a periodic function violates the periodicity behavior. Thus, taking into * Talk delivered by E. Ruiz Arriola 1 Note that they are not large in the topological sense, as discussed in [4,5]. account these Gribov replicas is equivalent to explicitly deal with genuine non-perturbative finite temperature gluonic degrees of freedom. A way of automatically taking into account LGT is by considering the Polyakov loop H as an independent variable, which in the Polyakov gauge becomes a diagonal unitary matrix H = eipA4(x) (3) invariant under the set of transformations given by Eq. (1). The relevance of the Polyakov loop in practical calculations is well recognized [1] but seldomly taken into account in high temperature calculations where large gauge invariance is manifestly broken since the gluon field is considered to be small. We have recently developed an expansion keeping these symmetries in general theories and applied it to QCD at the one quark+gluon loop level [6,7]. 2 The Center Symmetry In pure gluodynamics, or in the quenched approximation (valid for heavy quarks) at finite temperature there is actually a larger symmetry since one can extend the periodic transformations to aperiodic ones [3], g(x4 + |3) = zg(x4), zNc = 1 (4) so that z is an element of the center Z(Nc) of the group SU(Nc). This center symmetry is a symmetry of the action as well as the gluon field boundary conditions. An example of such a transformation in the Polyakov gauge is given by g(x4 ) = ei2nx4A/N P . (5) On the A4 component of the gluon field produces 2n (6) Thus, in the quenched approximation the period is N c times smaller than in full QCD. Under these transformations the gluonic action, measure and boundary conditions are invariant. The Polyakov loop, however, transforms as the fundamental representation of the Z(Nc) group, i.e. D —> zD, yielding (D) = z(D) and hence (D) = 0 in the unbroken center symmetry phase. At high temperatures one expects perturbation theory to hold, the gluon field amplitude becomes small and hence (D) —> 1, justifying the choice of D as an order parameter for a confinement-deconfinement phase transition. More generally, in the confining phase (Dn) = 0 for n = mNc (7) with m an arbitrary integer. The antiperiodic quark fields at the end of the Euclidean imaginary interval transform as q(x, (3) = — q(x,0) —> zq(x, (3) = —q(x,0), so that the center symmetry is explicitly broken by the presence of dynamical quarks. A direct consequence of such a property is that, in the quenched approximation non-local condensates fulfill a selection rule of the form, (q (np)q(0)) = 0 for n = mNc (8) since under the large aperiodic transformations given by Eq. (5) we have q (n|3)q (0) —> z-nq (np)q (0). This selection rule has some impact on chiral quark models. 3 Chiral quark models at finite temperature To fully appreciate the role played by the center symmetry in chiral quark models (for a recent review on such models see e.g. Ref. [8] and references therein) let us evaluate the chiral condensate at finite temperature. At the one loop level one has2 * = £(-1)n (10) n in terms of nonlocal Euclidean condensates at zero temperature. After Poisson resummation, at low temperatures we have °° TA A 2 * = 4 Note that in this formula (q (n|3)q(0)} refers to quarks uncoupled to the Polyakov loop while in Eq. (8) it refers to quenched QCD. at finite temperature in the presence of the Polyakov loop and in the chiral limit is given by fn2 = 4M2 T Trc^ (ft n d3 k 1 (2n)3 [¿^n + k2 + M2]2 The full calculation of the low energy constants at order O(p4) as a function of temperature and the Polyakov loop is carried out in Ref. [15]. The main feature is, similarly to (qq)* and f^, a strong suppression O(e-NcM|3) at low temperatures, but an enhancement of quark thermal effects close to the chiral-deconfinement phase transition. 6 Conclusions We see that the coupling of the Polyakov loop to chiral quark models at finite temperature accounts for large gauge invariance and modifies in a non-trivial way the results for physical observables. On the one hand, such a coupling allows to satisfy the requirements of chiral perturbation theory at low temperatures, generating a very strong suppression at low temperatures of quark loop effects. Nonetheless, the onset of deconfinement through a non vanishing value of the Polyakov loop accounts for a chiral phase transition at somewhat similar temperatures as in the original studies where the Polyakov loop was set to one. We expect this feature to hold also in the calculation of other observables. Although these arguments do not justify by themselves the application of these chiral quark-Polyakov models to finite temperature calculations, they do show that they do not contradict basic expectations of QCD at finite temperature. This work is supported in part by funds provided by the Spanish DGI with grant no. BMF2002-03218, Junta de Andalucía grant no. FM-225 and EURIDICE grant number HPRN-CT-2003-00311. References 1. D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. 53, 43 (1981). 2. N. P. Landsman and C. G. van Weert, Phys. Rept. 145 (1987) 141. 3. B. Svetitsky, Phys. Rept. 132 (1986) 1. 4. L. L. Salcedo, Nucl. Phys. B 549, 98 (1999) 5. C. Garcia-Recio and L. L. Salcedo, Phys. Rev. D 63, 045016 (2001) 6. E. Megias, E. Ruiz Arriola and L. L. Salcedo, Phys. Lett. B 563, 173 (2003) 7. E. Megias, E. Ruiz Arriola and L. L. Salcedo, Phys. Rev. D 69 (2004) 116003 8. E. Ruiz Arriola, Acta Phys. Polon. B 33 (2002) 4443 9. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 67, 074021 (2003) 10. J. Gasser and H. Leutwyler, Phys. Lett. B 184, 83 (1987). 11. W. Florkowski and W. Broniowski, Phys. Lett. B 386, 62 (1996) 12. A. Gocksch and M. Ogilvie, Phys. Rev. D 31, 877 (1985). 13. P. N. Meisinger, T. R. Miller and M. C. Ogilvie, Nucl. Phys. Proc. Suppl. 119, 511 (2003) 14. K. Fukushima, Phys. Lett. B 591 (2004) 277 15. E. Megias, E. Ruiz Arriola and L. L. Salcedo (in preparation) Application of chiral quark models to high-energy processes* Wojciech Broniowskia and Enrique Ruiz Arriolab Q The H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Cracow, Poland b Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain Abstract. We discuss the predictions of chiral quark models for basic pion properties entering high-energy processes: generalized parton distributions (GPD's) and unintegrated parton distributions (UPD's). We stress the role of the QCD evolution, necessary to compare the predictions to data. This is a very brief account of the talk based on Refs. [1-4], where the reader is referred to for the details and references. We discuss the use of low-energy chiral quark models to compute low-energy matrix elements of hadronic operators appearing in high-energy processes, in particular we evaluate the generalized and unintegrated parton distributions (GPD's and UPD's) of the pion in the Nambu-Jona-Lasinio model and the Spectral Quark Model [4]. We carry on the QCD evolution, necessary when comparing the model predictions to data obtained at much higher scales. The twist-2 GPD of the pion is defined as H(x,^-A2) dz i — z z — e^ --(n+(v')lq(0,-T>0)r+q(0>T>0)ln+(V)), where the quark operator q (z+, z-, z^ ) is on the light cone z2 = 0 and the link operators P exp(ig J"Z dx^A^) are implicitly present to ensure the gauge invariance (as usual we work in the light cone gauge A+ = 0). A similar definition holds for the gluon distribution. In chiral quark models the evaluation of H at the leading-Nc (one-loop) level is straightforward. For the NJL model with the Pauli-Villars regularization we get Hnjl (x, O,-Ai) 2 , , NcM2(1 — x)|Aj_| y- fsi + n — x)|Aj_|^ 1 + 2_ ci ® U-(1-x)|AjJ 1 = 01 -x)2Ai+4M2+4A2, e(x)e(i -x), s * Talk delivered by W. Broniowski where M is the constituent quark mass, At are the PV regulators, and ct are suitable constants. For the simplest twice-subtracted case, explored below, one has, for any regulated function f, the operational definition ^ctf(A?) = f(0) - f(A2)+ A2df(A2)/dA2. i We use M = 280 MeV and A = 871 MeV, which yields the pion decay constant fn = 93 MeV. In the SQM the result is 2 m2 (mp -(1 - x)2A2 ) H«*(*.o.-Ai> = ¿p;„'_xl4He|x|ii" - where mp is the mass of the p meson. We check that the pion electromagnetic form factor is FsqmW 1 mP dxHsQMlx,0,t) --^ mp +1' which is the built-in vector-meson dominance principle. For both models F(0) = 1 and Hsqm(x,0,0) = 6(x)6(1 - x). Our next goal is to compare the results to the data from transverse lattices [5]. We pass to the impact-parameter space via the Fourier-Bessel transformation, as well as carry the LO DGLAP perturbative QCD evolution from the low model scale Qo =313 MeV [6] up to the scale of the data. The results are shown in Fig. 1. We note that while the results at Q0 are completely different off the lattice data, when evolved to the scale of 500 MeV, corresponding to the lattice calculations, acquire a great resemblance to the data. In the second part of this talk we discuss the leading-twist UPD's of the pion, defined as q(x,ki) = dy d2ij± 16TT3 e-ixp+

A4M2Nc ^^ Injl(x,k_l, Qo) = —-—-——----20 X 0 1 -x) 4f2n3 (k2 + M2) (k2 + A2 + M2)2 C,» = ^ (2Ko,bM, - 2K0lb VOT, - ■ In SQM we find 6mp Isqm (x, k^, Qo) = —y--j7Tf5720W0(1 -x), n(k2 + mp/4)5/2 pNP r^^ / i , t^A^f ^ F^b) = (l+^)exp(-^) 0 transverse lattice, Q~500 MeV (S. Dalley) [i,j]: bx=ib0, by=jb0, b0~2/3 fm ,_, 0. 8 X _Q 0. 6 > 0 4 0. 2 0.2 0.4 0.6 0. 8 „ 1 Fig. 1. GPD of the pion in the impact-parameter space plotted as a function of the Bjorken x. Top: model for four momentum scales, from 313 MeV up to 2 GeV. Bottom: transverse lattice [5]. Numbers in brackets label the plaquette [1]. The qualitative agreement to the data is achieved at the scale of about 500 MeV. (the meaning of b different here, it is the transverse coordinate conjugated to k2). The above results are at the low model scale Qo. Next, we evolve these UPD's from Q0 to high scales with the Kwiecinski equations [2], obtained in the CCFM framework. The results are displayed in Fig. 2. One may show several qualitative and quantitative results concerning UPD's. At large b they fall off exponentially and at large k2 they fall off as a power law. Spreading with increasing Q2 occurs, with (k2 ) ~ Q2aS (Q2). Also, asymptotic formulas at limiting cases may be explicitly given [2] which may be useful in checking numerical calculations of CCFM-type cascades [7]. Our basic conclusion is that chiral quark models may be used to provide GPD's and UPD's (also the pion distribution amplitude [3] not presented here) at the low model scale, Qo. Upon evolution to higher scales, the agreement with the data (experimental or lattice) is very reasonable. Fig. 2. Valence quarks (dashed lines), sea quarks (dotted lines), and gluons (solid lines), for the transverse coordinate b = 0,1,2, 3,4,5, and 10 fm (bottom to top). Evolution with the Kwiecinski equations from the model scale Q o=313 MeV up to Q = 2 GeV has been made. References 1. W. Broniowski and E. Ruiz Arriola, Phys. Lett. B 574, 57 (2003) [arXiv:hep-ph/0307198]. 2. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 70, 034012 (2004) [arXiv:hep-ph/0404008]. 3. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 66, 094016 (2002) [arXiv:hep-ph/0207266]. 4. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 67, 074021 (2003) [arXiv:hep-ph/0301202]. 5. S. Dalley, Phys. Lett. B 570,191 (2003) [arXiv:hep-ph/0306121]. 6. E. Ruiz Arriola, lectures given at 42nd Cracow School of Theoretical Physics, Flavor Dynamics, Zakopane, Poland, 31 May - 9 Jun 2002, Acta Phys. Polon. B 33 (2002) 4443. 7. H. Jung, Comput. Phys. Commun. 143, 100 (2002) [arXiv:hep-ph/0109102]. On the Meson-Few-Body Problem From a Few-Nucleon Perspective* Luciano Cantona and Leonid G. Levchukb Q Instituto Nazionale di Fisica Nucleare, 35131 Padova, via F. Marzolo, n. 8, Italy b NSC Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine Abstract. The phenomenology of pion production from nucleon-deuteron collisions is analyzed, with reference to the outgoing channel where the three-nucleon system is bound. The available experimental data, from threshold up to the A resonance, are compared with calculations using accurate nuclear wavefunctions coming from rigorous solutions of the three-nucleon quantum mechanical equations. The dominant contributions for pion production are obtained through matrix elements involving pion-nucleon rescattering mechanisms in S- and P-waves. S-wave rescattering includes also an isoscalar contribution which is generally suppressed for low-energy pion-nucleon scattering, but becomes enhanced for pion production because the latter implies a different kinematical regime, which involves high-momenta contributions. P-wave rescattering includes also explicitly the A degrees of freedom. It is found that the existing data could be described reasonably well with enhanced S-wave rescattering in the isospin-even channel as is described by the Hamilton model. Initial-state interactions (ISI) between the proton and the deuteron have, in general, sizable effects on the spin-averaged and spin-dependent observables. These ISI effects become very important for spin observables involving interference terms amongst the various helicity amplitudes, such as for the nucleon vector analyzing power Ay . The study of pion production from nucleon-deuteron collisions (this reaction is called also "pionic capture of nucleons on deuterons") represents an interesting topic of research. Potentially, it interconnects low-energy few-nucleon physics with intermediate-energy physics, pion dynamics, etc. With this reaction it is possible to study the NN —> NNn inelasticities in the most simple (complex) nuclear environment, the three-nucleon system, where rigorous few-body techniques have been developed to describe adequately the nucleon dynamics. But these reactions can also represent a window, independent and complementary, to the diagrams that presumably contribute to the three-nucleon forces. Traditionally, 3NF's are constructed phenomenologically in low-energy few-nucleon physics to overcome some deficiencies in the three-nucleon and more-nucleon systems, with parameters adjusted ad hoc to reproduce some data that could not be reproduced with a given set of conventional 2N potentials. It is most desirable that those diagrams contributing to the 3NF can be studied independently * Talk delivered by L. Canton by other experiments, and pion production reactions could be the kind of process that might shed light on these diagrams. Another general aspect that makes pion-capture reactions quite interesting is the dependence of the associated amplitude upon the distribution of the nucleonic axial currents. This makes these processes closely related to neutrino reactions in nuclei, which is another important topic of research in present days. d V 0 N- N p,n . n N Fig. 1. Diagrammatic representation of the calculation required for determining the pion production amplitude from nucleon-deuteron collisions. Left, the overall diagram. Right, the elementary production mechanisms. A N N n N N N n n P N N N N n N N N N In Fig.1 the base calculation for pionic capture is illustrated. It involves use of accurate 3N bound-state wavefunctions, calculation of ISI via Faddeev-Alt-Grassberger-Sandhas techniques [1], and an exhorbitant number of multidimensional integrals for the partial-wave evaluation of the elementary pion-production processes. Details and updates for the present calculations can be found in Refs. [2]. The elementary production mechanisms (shown in the r.h.s. of Fig.1) are calculated starting from the phenomenological low-energy interaction Lagrangian, coupling the pion with the nucleon field £ int ■f \ \ a o An—^^j^ ■ [® x ®] - 47t-2-W [® • ®] m„ mn m„ (1) and with the A field fnN A at, (¥ A T V ■ a^O + h.i (2) The calculations herein illustrated have been performed with the following set of parameters: fnNN/4n = 0.0735, f^A^n = 0.32, Ar = 0.045 and AO = 0.006. A crucial aspect is represented by the off-shell extrapolations of these parameters in the evaluation of the n-production matrix elements. For the S-waves terms we have (t is the square of the four-momentum transfer of the corresponding exchange particle) AOFF :A?N- AP t AP OFF AO NaSR + aff^ asR + ac (3) m n 2 2 a m t P O 2 t P The form on the left denotes the isospin-odd contribution in terms of a p-exchange model, while on the r.h.s. we describe the isospin-even term as the combined effect of phenomenological short-range (SR) processes and an effective scalar-meson (ct) exchange. The two combined effects act in opposite directions [3]. The form on the right leads to an off-shell enhancement of the probability amplitude for pion production in the scalar-isoscalar channel. The NN —> NNn inelasticities have been studied extensively in the case of the simpler reactions pp —> n+d; pp —> ppno; and pp —> n+pn. Reference to earlier works can be found in [4]; a more updated review is Ref. [5]. An interesting element of debate concerns the possible mechanisms responsible for the production yield for the process pp —> ppno at threshold. This yield has been explained by resorting to two different mechanisms, represented in Fig. 2. To ascertain which is the kind of mechanism that contribute most likely to the production process remains still an open issue. Studies performed in pion production from nucleon-nucleon collisions have been able to exclude neither of the two mechanisms shown in Fig. 2. Fig. 2. Alternative mechanisms suggested for the pp —> ppno reaction at threshold. Left, pion coupling to short-range two-body exchange currents, Ref. [6]. Right, pion rescattering in the isoscalar channel, Ref. [7]. One hopes that the reactions pd —> no 3 He and pd —> n+ 3 H could help to clarify the question of which is, if any, the correct mechanism that describe the process in the isospin even channel. These three-nucleon-type reactions are extremely complicated, and therefore much more difficult to analyze theoretically. On the other hand, here the interference effects amongst the various mechanisms are much more important than for two-nucleon collisions at threshold and therefore these reactions might represent a more stringent test for the possible mechanisms that describe the process. In the following, I will present results obtained assuming that the production process in the scalar-isoscalar channel is dominated by the rescattering model (the two mechanisms on the r.h.s of Fig. 2); the mechanism depicted on the l.h.s of the same figure will be possibly analyzed in the future. The current (spin-averaged) experimental situation has been greatly improved after the addition of recent Cosy data, as exhibited in Fig. 3. 100 90 80 70 2T 60 i ~ 50 s^ *o 40 30 20 10 0.5 1 1.5 2 2.5 n Fig. 3. The current experimental situation for the excitation function, in the energy range from threshold up to the A-resonance, after adding recent Cosy data (black diamonds and triangles). Reference to data can be found in [8]. The parameter n represents the pion c.m. momentum in units of pion masses. The next figure, Fig. 4, shows the excitation function for various plane-wave calculations, using the Bonn-B model for 2N interaction. The dotted line denotes calculations with the standard (nonrelativistic) definition for the nNNN Jacobi momenta. The other two lines refer to the results with replacement of the pion mass by its total energy in this Jacobi momentum set (see details in Ref. [10]). The dotted-dashed line denotes calculations where pion rescattering in the scalar-isoscalar channel is omitted, while the solid line accounts for our full model (which includes the mechanisms on the r.h.s. of Fig. 2). Fig. 5 shows on the l.h.s. the differential cross-section in collinear kinematics for the pd —> no 3 He process. Calculations are for various 2N potentials. It is seen that ISI have a significant effect on the angular dependence of the differential cross-section, in particular at backward angles. The same figure shows on the r.h.s the dramatic effect that ISI's have on the proton analyzing powers Ay. In the region of interference between s-wave and p-wave mechanisms, which corresponds approximately to n ~ 0.5, the structure of Ay exhibits a rapid variation in sign, with the appearance of an additional "bump" in the angular distribution. This structure is reproduced by our complete model independently of the selected 2N potentials, once the effects of ISI are taken into account. -e—i Nikulin et al. -■—1 Pickar et al. COSY (?He+n0) ' COSY (3H+n+) -s—1 older data ** ▼t I* iT [] m n Fig. 4. Excitation function for various plane-wave calculations, using the Bonn-B model for 2N interaction. See text for details on the line notation. 0 -0.4 0.01 0.1 1 o 90 180 n e Fig. 5. (Left) Differential production cross-section in collinear kinematics. (Right) Proton analyzing power in the "interference" region between p-wave and s-wave mechanisms. In all panels, the thin solid line denotes plane-wave calculations with Bonn-B interaction. The thick lines denote ISI calculations for different 2N potentials, dashed (Bonn A), solid (Bonn-B), dotted (Paris). Another interesting observable that has been analyzed is the deuteron tensor analyzing power T20 (Fig. 6). The production reaction acts at threshold as an helicity selector, in that the observed T20 is close to its geometrical limit -y/2. This limit can be obtained in plane-wave calculations with pure isovector nN s-wave rescattering [9]. However, the trend with energy is much better reproduced once the nN rescattering in the scalar-isoscalar channel are also considered. The reproduction improves further once ISI are taken into account [10]. 0.2 o 11 CD -0.2 - -0.4 - -0.6 - -0.8 - -1 -1.2 -1.4 -1.6 0.01 0.1 10 n Fig. 6. Deuteron tensor analyzing power T20 in (forward) collinear kinematics. All lines refer to plane-wave calculations. The solid line denotes the full model. The dot-dashed line refers to calculations without s-wave nN rescatterings in the scalar-isoscalar channel, the dotted line to calculations deprived also of s-wave rescatterings in the isovector channel. 0 1 In conclusion, I have reported on progresses made recently on the pion-3N system, with respect to the pion-production reaction pd —> no 3 He. The phenomenology of this reaction is quite complicated, especially if one starts to consider the spin-structure of the process. It is shown that a fair understanding of the process is possible, provided that the variety of elementary production mechanisms used to descibe pion-production from 2N collisions are taken into account, and the nuclear 3-body aspects (bound-states and scattering effects in the initial state) are carefully calculated. Also the interference effects amongst the various production mechanisms are quite important in these processes. This should draw attention to these type of reactions: they could help to clarify the issue of the nature of the pion-production mechanisms for the process pp —> ppno at threshold. Acknowledgment I would like to thank the warm hospitality of the organizers of the mini-workshop on quark dynamics in Bled. I express also many thanks to T. Melde, G. Pisent, W. Schadow, A. Shebeko, and J.P. Svenne, for sharing interest and collaborating on many topics that are of relevance for this subject. This work has been co-funded by the Italian MIUR-PRIN project "Fisica Teorica del Nucleo e dei Sistemi a Molti Corpi". L. Levchuk thanks the University of Padova and the INFN for the support and hospitality. References 1. E.O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B2,167 (1967). 2. L. Canton and W. Schadow, Phys. Rev. C 56, 1231 (1997); L. Canton et al. Phys. Rev. C 57,1588 (1998); L. Canton et al. Nucl. Phys. A684, 417c (2001); L. Canton and W. Schadow, Phys. Rev. C 61, 064009 (2000); L. Canton and L. Levchuck, submitted for publication. 3. J. Hamilton, High Energy Physics, E.H.S. Burhop ed. (Academic, New York, 1967); O.V. Maxwell, W. Weise, and M. Brack, Nucl. Phys. A348, 338 (1980). 4. H. Garcilazo and T. Mizutani, nNN Systems World Scientific, Singapore (1990). 5. C. Hanart, Phys. Rept. 397,155 (2004) 6. T.S-H. Lee and D.O. Riska, Phys. Rev. Lett 70, 2237 (1993). 7. E. Hernandez and E. Oset, Phys. Lett. B350 158, (1995). 8. C. Kerboul et al. Phys. Lett. B 181, 28 (1986); V. Nikulin et al. Phys. Rev. C 54, 1732 (1996); M.A. Pickar et al. Phys. Rev. C 46 397 (1993); M. Betigeri et al. Nucl. Phys. A690 473 (2001); S. Abdel-Samad et al. Phys. Lett. B553, 31 (2003). 9. J.F. Germond and C. Wilkin, J. Phys. G 16, 381 (1990). 10. L. Canton et al. in preparation. Quark matter and quark stars* M. Fiolhaisa, L. P. Linaresb, M. Malheirob, A. Taurinesc Q Departamento de Física and Centro de Física Computational, Universidade de Coimbra, P-3004-516 Coimbra, Portugal b Instituto de Física, Universidade Federal Fluminense, Av. Litoranea, 24210-310 Niteroi, Brazil c Instituto de Física Teórica, Universidade Estadual Paulista, 01405-900 Sao Paulo, Brazil Abstract. Strange quark matter is studied in the framework of the Chromodielectric model (CDM), and the corresponding equations of state are used to investigate the structure (mass and radius) of cosmological compact objects. At high densities, the phase of QCD known as "Color Flavour Locked" (CFL) phase, may also be modelled in the CDM through the inclusion of a direct quark-quark correlation (pairing) energy. We studied this phase, obtaining the corresponding equation of state, which we compared with the equations of state of the other phases. The results show that the equations of state obtained in the framework of the CDM are similar to those obtained in the framework of QCD. On the other hand, the CFL phase turns out to be more stable than the normal (beta equilibrium) phase. Work is still in progress regarding the structure of quark stars which are obtained using the equation of state for the CFL phase. The chromodielectric model is an effective model for the interactions amongst quarks in the low and intermediate energy range [1]. The interactions between the quarks are mediated by meson exchange: the scalar-isoscalar sigma (ct), the pseudoscalar-isovector pion (n), and the scalar, isoscalar, chiral singlet chi (x), which is the responsible for generating confinement in the model (the x field is usually viewed as a glueball field). The chromodielectric model is chiral symmetric, a symmetry that is spontaneously broken, leading to a dynamical generation of quark masses which depend on x-1. The mesons experience self-interactions: a 'mexican-hat' potential for the chiral mesons and, for the confining field, a potential which is usually written in the form [2] where mx is the x mass, and y and n are parameters. It has a global minimum at X = 0 and a local one at x = Ymx. The height of the local minimum, U(ymx) = (nmx)4 = B, is interpreted as a "bag pressure" [3], as in the MIT bag model, and this correspondence is used to fix the model parameters. * Talk delivered by M. Fiolhais (1) The chiral CDM has been applied in studies of the baryon structure, in particular, the nucleon and the Delta resonance [4] and the Roper [5]. These states are described as chiral solitons, with three valence quarks dynamically confined to a bag whose radius is approximately 1 fm. Quark matter has also been studied in the framework of the model [6]. The simplest system which can be considered in this context is the charge neutral u and d infinite quark matter, to which a semiclassical (Thomas-Fermi) formalism can be applied. Such treatment allows us to readily obtain the Equation of State (EOS) for the quark matter. In a recent study [7], we considered strange quark matter which, in addition to u and d quarks, also contains the strange quark, s. The requirement of (local) charge neutrality and beta equilibrium enforces the need to include electrons into the system. In the semiclassical approximation (quarks described by plane waves, and constant classical meson fields) we found two distinct stable solutions for the same set of model parameters. In both cases, ct remains always close to fn irrespective of the density. In one solution (solution I in [7]), the x field stays close to zero, being a slowly increasing function of the density. For such small x, the quar-tic potential of the CDM is indistinguishable from U = , thus, in practice, this solution I corresponds to the one obtained and used by Drago et al. [8] in the framework of the simpler quadratic potential [y —> oo in the potential (1)]. Due to the smallness of the x field, quark masses are large and the system is in a chiral broken phase. As a consequence of the charge neutrality and beta equilibrium, which are imposed in the variational calculations, a certain number of electrons are required in the system which contains, on the other hand, u and d quarks (in almost the same abundance) and s quark (less abundant than the other flavours). There is another solution (solution II in [7]) corresponding to a large confining field. This solution cannot exist for quadratic potentials such as the one in [8] and it corresponds to a chiral symmetric phase: the quark masses for the three flavours are all very close to zero, almost independent of the density. The chemical potentials (m) for each flavour are dominated by the Fermi momentum and one has — |Md — Ms. Hence, in this chiral symmetric phase, which pertain to the quartic potentials of the type (1), the abundance of quarks u, d and s are the same and there are no electrons, i.e. — 0. Our results in Ref. [7] indicate that strange quark matter at high densities (solution II) is not absolutely stable. However, it is expected that a new phase that is supposed to occur in QCD at very high densities, known as color flavor locked (CFL) phase [9], is likely to be the ground state even if the quark masses are different from each other [10]. This suggests that the strange matter described by our solution II in Ref. [7] may undergo a transition to the CFL phase, whose energy is lowered due to a quark BCS-like pairing interaction. A recent study described how the CFL phase in dense matter enhances the parameters space for absolute stability of the strange matter [11]. In that study, a phenomenological vacuum energy density or bag constant B is included in the spirit of the MIT bag model. It was shown that, when the gap energy of the QCD Cooper pairs increases, the bag constant can be larger and the strange matter is absolutely stable (i.e., the energy per particle is lower than for the iron). We performed a similar study in the CDM model including the BCS quark pairing in it and analyze the superconducting color flavor locked phase. The quark pairing interaction is introduced through the inclusion of the extra term 3A2 p.2/n2 in the energy density. The (adjustable) parameter A is the paring energy and | is the (quark) chemical potential. Our goal was to investigate whether the inclusion in the energy density of a negative term of the diquark condensate would maintain the stability of quark matter even for a large potential energy. The calculation was carried on using the parameters y = 0.2 MeV, n = 0.096 and mx = 1.7 GeV for the U(x) potential. In Fig. 1 we plot the energy per particle as a function of the density (M = 939 MeV) to study the stability in the CFL for CDM (solution II in Ref. [7]) and QCD. We present a comparison between the EOS of the CDM + CFL and QCD in the CFL, for A = 100 MeV and A = 0 (no pairing interaction). The results show that the EOS obtained for the CFL strange matter in the CDM is very similar to the QCD one, suggesting that the CDM has the most relevant features of QCD at high densities. Increasing the quark pairing A interaction, the strange quark matter becomes more stable. This result indicates that even for large potential energies of the confining field (n > 0.096) the CFL strange quark matter may exist as an absolutely stable state. 300.0 200.0 -100.0 -200.0 0.0 0.2 0.4 0.6 0.8 1.0 P [fm3] Fig.1. Stability of the strange quark matter in the CFL phase for CDM (solution II in Ref. [7]) and QCD [11]. This study can be relevant for the structure and formation of compact quark [12,13] and hybrid stars [14]. In Ref. [7] we showed that quark stars with one solar mass and a radius in the range 6-8 km could be formed if no pairing interaction was included, though they were not absolutely stable. We expect that CFL stars might be absolutely stable and even more compact than strange stars. References 1. H. B. Nielsen, A. Patkos, Nucl. Phys. B 195, 137 (1982); H. J. Pirner, Prog. Part. Nucl. Phys., 29, 33 (1992) 2. M. C. Birse, Prog. Part. Nucl. Phys., 25,1 (1990) 3. M. Rosina, A. Schuh, H. J. Pirner, Nucl. Phys. A448, 557 (1986) 4. T. Neuber, M. Fiolhais, K. Goeke, J. N. Urbano, Nucl. Phys. A 560, 909 (1993); A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 609, 488 (1996) 5. P. Alberto, M. Fiolhais, B. Golli, J. Marques, Phys. Lett. B 523, 273 (2001) 6. A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 588, 801 (1995); W. Broniowski, M. Cibej, M. Kutschera, M. Rosina, Phys. Rev. D 41, 285 (1990); J. A. McGovern, M. Birse, Nucl. Phys. A 506, 367 (1990); Nucl. Phys. 506, 392 (1990) 7. M. Malheiro, M. Fiolhais and A. R. Taurines, J. Phys. G29,1045 (2003) 8. A. Drago, U. Tambini, M. Hjorth-Jensen, Phys. Lett. B 380,13 (1996); Prog. Part. Nucl. Phys. 36, 407 (1996) 9. M. Alford, K. Rajagopal and F. Wilczek Nucl. Phys. B537, 433 (1999) 10. M. Alford, J. Berges and K. Rajagopal Nucl. Phys. B558, 219 (1999) 11. G. Lugones, J. E. Horvath Phys. Rev. D66, 074017 (2002) 12. G. Lugones, J. E. Horvath Astronomy and Astrophysics 403,173 (2003) 13. A. Drago, A. Lavagno and G. Pagliara, nucl-th/0401052. 14. P.K. Panda, D.P. Menezes and C. Providencia Phys. Rev. C69, 025207 (2004) A Time Dependence of the Scale of QCD H. Fritzsch Sektion Physik, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany Abstract. From astrophysics there are indications that the finestructure constant a has changed during the past 10 billion years. Within grand unification one can deduce that also the QCD scale has changed. Tests for a time variation of this scale are described. The result of the new experiment in Munich is discussed. The theory of QCD is very remarkable. It is a theory of very few parameters, i.e. only Ac and the quark masses. The latter are related to inputs by the flavor interactions and have nothing to do with the strong interactions. The parameter Ac just sets the scale of the strong interactions and is not a real parameter for the strong interaction itself. Thus the QCD-theory, proposed by Gell-Mann and myself in 19721, is indeed an exceptional theory, describing lots of complexities in terms of very few parameters, which, as discussed below, might even depend on time. Usually in physics, especially in particle physics, we deal with the local laws of nature, say the field equations of QCD or the Maxwell equations. But when it comes to the fundamental constants, like the finestructure constant a, we must keep in mind that also questions about the boundary conditions of the universe come up. We do not know, where these constants, like a or as or the lepton and quark masses, come from, but it could well be that at least a few of them are products of the Big Bang. If the Bing Bang would be repeated, these constants could easily take different values. But in this case it is clear that the constants could never be calculated. So in connection to the fundamental constants the question comes up, whether they are really cosmic accidents, or whether they are determined by the dynamics, whether they are changing in time or in space, or whether they are indeed calculable in a hypothetical theory going far above the present Standard Model. Also considerations related to the Anthropic Principle should be made. Life in our universe can exist only if the values of the fundamental constants take on certain values. In a universe in which, for example, the u-quark is heavier than the d-quark, the proton would decay in a neutron, and life would not exist, at least not in a form known to us. Of course, today a is just the interaction constant, describing e. g. electron-scattering at low energies: a-1 = 137.03599976. (1) But it is remarkable. Based on this number, one can calculate all effects in QED to an accuracy of about 1 : 10.000.000, e. g. the magnetic moment of the electron. Of course, QED is only a part of the Standard Model of today, based on a superposition of QCD and the SU(2) x U(1) - electroweak theory, and a is just one of at least 18 parameters, entering the Standard Model. One of the fundamental quantities is the proton mass. I should like to stress that the proton mass is a rather complicated object in the Standard Model. The coupling constant of QCD follows in leading order the equation: Here the scale parameter A enters, which has been determined to be: A = 214+35 MeV. (3) A is a free parameter of QCD, and all numbers of QCD scale with A, at least in the limit where the masses of the quarks are set to zero. But A can be expressed in terms of MeV, i. e. it is given in reference to the electron mass, which is outside QCD. The physical parameters like the proton mass are simply proportional to A, apart from a small correction due to quark masses. The scale of confinement of the quarks is inversely proportional to A. I should also remind you that Grand Unification imposes that the parameters as, a and aw are not independent. They are related to each other, and related to the unified coupling constant, describing the interaction at the unification scale Aun. It is known that the group SU(5) does not describe the observations, since the three coupling constants do not converge precisely. If supersymmetric particles are added at an energy scale of about 1 TeV, a convergence takes place, however2. In SO(10), proposed by P. Minkowski and me3 the situation is different, since in this group the unification is a two-step process, where another mass scale, the mass scale for the righthanded W-boson, enters. If this mass scale is chosen in the right way, the unification can be achieved without supersymmetry. After these preparations let me come to the question of time dependence. A group of physicists has recently published their evidence that the finestructure constant had a different value billions of years ago4. They were investigating the light from about 134 quasars, using the so-called "many multiplet method". They were looking at the fine-structure of atomic lines, originating from elements like Fe, Ni, Mg, Sn, Ag etc. One particular aspect is that the fine-structure is a rather complex phenomenon, fluctuating in particular also in the sign of the effect. These sign changes have been observed and used in fixing the experimental values of a. The result is: Aa — = (-0.72 ±0.18) • 10~5 . (4) Thus a was slightly larger in the past. If one takes a linear approximation and uses a cosmic lifetime of 14 billion years, the effect is a/a « 1.2 ■ 10-15 per year. If a depends on time, the question arises, how this time-variation is generated. Since a = e2/fie, a time variation could come from a time variation of fi or e. Both cases are, I think, not very likely. If e depends on time, it would mean, that we have a serious problem with relativity. If fi would depend on time, atomic physics runs into a problem. So I think that a time dependence of a simply means that e is becoming time-dependent. Let me also mention that according to the results of Dyson and Damour5 there is a rather strong constraint on a time-variation of a, derived from the investigation of the remains of the Oklo reactor in Gabon. If no other parameters change as well, the relative change (a/a) per year cannot be more than 10-17, i. e. there is a problem with the astrophysical measurements, unless the rate of change for a has become less during the last 2 billion years. The constraint is derived by looking at the position of a nuclear resonance in Samarium, which cannot have changed much during the last 2 billion years. However, I tend not to take this constraint very seriously. According to the Grand Unification as and A should have changed as well, and the two effects (change of a and of A) might partially cancel each other. The idea of Grand Unification implies that the gauge group SU(3) of the strong interactions and the gauge group SU(2) x U(1) of the electroweak sector are subgroups of a simple group, which causes the unification. Both the groups SU(5) and S0(10) are considered in this way. I like to emphasize that the group SO(10) has the nice property that all leptons and quarks of one generation are described by one representation, the 16-representation. It includes a righthanded neutrino, which does not contribute to the normal weak interaction, but it is essential for the appearance of a mass of the neutrino, which is expected in the S0(10)-Theory. In SU(5) two representations of the group are needed to describe the leptons and quarks of one generation, a 10- and a (5 )-representation. I should also like to emphasize that the gauge couplings a$, aw and a meet in the SU(5)-theory only, if one assumes that above about 1 TeV supersymme-try is realized. In the S0(10)-theory this is not needed. A convergence of the coupling constants can be achieved, since at high energies another energy scale enters, which has to be chosen in a suitable manner. A change in time of a can be obtained in two different ways. Either the coupling constant aun stays invariant or the unification scale changes. I consider both effects in the SU(5)-model with supersymmetry. In this model the relative changes are related: 1a 8 1 10 Aun (5) --=a--"^A— (5) a a 3 as n Aun One may consider the following scenarios: 1) Ag invariant, au = au(t). This is the case considered in6 (see also7), and one finds and 1 a 8 1 et s a a 3 as as A „„ = i47taeff(-k2) D^e(k) [— Y^leg [T Y^Hf , (1) where e, f, g, h schematically represent spinor, color and flavor indices, aeff (—k2) = aeff(Q2) denotes an effective running coupling, and D^f (k) is the free gluon propagator in the gauge in which the aforementioned SD studies have been carried out almost exclusively, namely the Landau gauge: = + (2) The BS equation for the bound-state vertex rqq / of the meson composed of the quark of the flavor q and antiquark of the flavor q', is then [rqq / (k,P)]ef d4 o p p The consistent RLA requires that the same interaction kernel (1) be previously used in the SD equation for the full quark propagator Sq. That is, dressed quark propagators Sq (k) for various flavors q, S-1 (p)= Aq(p2)P - Bq(p2) , (q = u,d,s,...) , (4) are obtained by solving the gap SD equation Sq 1 (p) = P- Taq - i4n d40 r , Aa „„ ,„,Ab aeff[-(P -«)2]ö£(p -l)o -r Y^Sq (i)—r , (5) (2n)4 eiIL ; J 2 ' ^ ' 2 where mq is the bare mass of the quark flavor q breaking the chiral symmetry explicitly. The case mmq = 0 corresponds to the chiral limit where the current quark Fig. 1. A typical example of a momentum-dependent dynamically generated constituent quark mass function, which has values of the order of one-third of the nucleon mass at low momenta squared. It is the non-strange (q = u) dynamical quark mass of the lightest quark flavor, Mu(—Q2), calculated using the effective coupling (18) and the input parameters given in Ref. [14] by Eq. (26) there. In the exact chiral limit, the result for Mu ( —Q2) is very similar. mass mq = 0. In that limit the dynamically generated "momentum-dependent constituent mass function" Mq (p2) = Bq (p2)/Aq (p2) stems exclusively from the nonperturbative phenomenon of DxSB. (Of course, for any m.q which is small with respect to the typical hadronic mass scale ~ 1 GeV, Mq(p2) stems largely from DxSB for values of p2 below the perturbative domain.) The "constituent quark mass" can be defined as the value of this momentum-dependent constituent mass function at some low —p2, say p2 = 0. The important thing for obtaining a successful hadronic phenomenology, especially in the light-quark sector (q = u, d, s), is that DxSB be sufficiently strong; i.e., the gap equation (5) should yield quark propagator solutions Aq(p2) and Bq(p2) giving the dressed-quark mass function Mq (p2) whose values at low —p2 are of the order of typical constituent mass values, namely several hundred MeV, as exemplified in Fig. 1. It turns out that the interaction (1), or, equivalently, aeff(Q2), which would lead to successful hadronic phenomenology through RLA, must have fairly high integrated strength in the domain of intermediate (around Q2 ~ 0.5 GeV2) and low momenta. Only then RLA equations (4) and (3) can give acceptable description of hadrons, notably mass spectra and DxSB [20,24]. On the other hand, at large spacelike momenta, the running coupling aeff (Q2) must reduce to apert(Q2), the well-known running coupling of perturbative QCD. The question is how to obtain theoretically such an interaction. Fig. 2. The effective coupling (18) proposed and analyzed in the present paper is depicted by the solid curve. The two older and also phenomenologically successful effective strong running couplings, namely JM [21] and MRT [23,20] aeff (Q2 ), are depicted by, respectively, dashed and dash-dotted curves. 3 Strong coupling enhanced by gluon condensates We already gave an answer to the above question in our paper [14]. There, we pointed out that such an interaction kernel for SD studies in RLA and the Landau gauge resulted from combining the form the running coupling has in the Landaugauge SD studies, namely Eq. (6) below [19,25-28], and the ideas on the possible relevance of the dimension 2 gluon condensate (A2) [7-13,3-6]. In the present paper, we give a simplified and more intuitive derivation thereof as follows. The definition of the strong running coupling as(Q2) which is appropriate for the SD studies in the Landau gauge [19,25-28] is as(Q2) = as(H2) Z(Q2) G(Q2)2 , (6) where Os(m2) = g2/4n and Z(m2)G(m2)2 = 1 at the renormalization point Q2 = M2. The gluon renormalization function Z(-k2) defines the full gluon propagator D^v(k) in the Landau gauge: D^(k) = Z(-k2)D^(k) = + . (7) Similarly, G(-k2) is the ghost renormalization function which defines the full ghost propagator: DG(k) = ^l. (8) It is sometimes convenient to express the gluon and ghost renormalization functions through the corresponding gluon (A) and ghost (G) polarization functions nc (Q2) and nA(Q2): 11 Z(Q ) = - = (9) I i- Q2 I i- Q2 From the recent flurry of papers on (A2), Refs. [7-10] are particularly relevant for the exposition below, but already a long time ago it was pointed out by, e.g., Refs. [3-6], that in the operator product expansion (OPE) the gluon condensate (A2) can contribute to QCD propagators. Their [3-6] (A2)-contributions to the OPE-improved gluon (A) and ghost (G) polarization functions were recently confirmed by Kondo [10]. For LG (adopted throughout this paper), number of QCD colors Nc = 3 and space-time dimensions D = 4, their expressions for the polarizations reduce to ni(Q2)= m2 + Oi(1/Q2), (i = A, G), (10) 3 ™i = 32 92 1) were not considered at all. Thus, at this point, the only practical approach is that the contributions Oi(1/Q2) (i = A, G) in Eq. (10) are approximated by the terms « 1/Q2 and parametrized, i.e., C C OaO/Q2)«-^, og{ 1/Q2)«q|. (12) In Eqs. (12), both CA and CG would thus be free parameters to be fixed by phenomenology. Still, we should mention that the effective gluon propagator advocated by Lavelle [29] would indicate CA « (0.640 GeV)4 for the following reason: for LG and D = 4, the contribution which this gluon propagator receives from the so-called "pinch diagrams" vanishes, and its [29] OA(1/Q2) polarization 2 _ 34Nc7t«s(F2) _ (0.640 GeV)4 "A ^ 9(N2-1)Q2 ~ Q2 1 j originates entirely from the gluon polarization function in Ref. [4], provided one invokes some fairly plausible assumptions, like using equations of motion, to eliminate all condensates except (F2). (The quark condensate (qq) could also be neglected [29].) Since Ref. [30] indicates that the true value of as (F2) is still rather uncertain, and since Refs. [5,6] make clear that Lavelle's [29] propagator misses some (unknown) three- and four-gluon contributions, we do not attach too much importance to the precise value CA = (0.640 GeV)4 [29,1] in Eq. (13), but just use it as an inspired initial estimate. Fortunately, the corresponding variations of CA still permit good phenomenological fits, since we found in Refs. [14,18] that our results are not very sensitive to CA. We do not have a similar estimate for CG, but one may suppose that it would not differ from CA by orders of magnitude. We thus try CG = CA = (0.640 GeV)4 as an initial guess. It turns out, a posteriori, that this value of CG leads to a very good fit to phenomenology [14,18]. We can now give a general, although heuristic argument why the contribution (11) of the dimension-2 (A2) condensate to the gluon and ghost polarizations (10), should indeed lead to the form of aeff(Q2) already found in Ref. [14]. As the first step, let us assume that in the gluon and ghost polarization functions, nA and nG, one can disentangle the perturbative (pert) from nonperturbative (Npert) parts, n = npert + nNpert (i = A, G). At least for high momenta Q2, it is then possible to approximately factor away the perturbative from nonperturbative contributions; for i = A, Z(Q2)«-¿s—--¿s-= Zpert(Q2) ZNpert(Q2), (14) i i nfr (Q2) i | (Q2) I H Qi I H--q7- where we neglect the nAert(Q2)nApert(Q2)/Q4 term. Analogously, G(Q2) «-L--nL-= Gpert(Q2) GNpert(Q2). (15) i i nper (Q2) i | n^p (Q2) Q2 n Q2 For sufficiently high Q2, the general QCD coupling as (Q2) reduces to the perturbative QCD coupling apert(Q2), so that Eq. (6) suggests that (g2/4n) Zpert(Q2) Gpert(Q2)2 should be identified with apert(Q2). For high Q2, we can also assume that nonperturbative parts are given by the OPE-based results of Refs. [3-6,10] [which in our present case boil down to Eqs. (10)-(11)], and by the parametrization (12). Then ZNpert(Q2) =-J r , (16) 1 I mA I CA ' "T" Q2 "T" Q4 GNpert(Q2) =-J r , (17) 1 _ mA I CG -qT-^QT where mA is given (11) by the (A2) condensate. Obviously, Eqs. (6), (14), (15), (16) and (17) suggest an effective coupling aeff(Q2) of the form aeff(Q2) = apert(Q2) ZNpert(Q2) GNpert(Q2)2 . (18) The derivation of the coupling (18) is obviously only heuristic and is far from rigorous. However, we already obtained the same result through a more rigorous derivation in Ref. [14]. Already in Ref. [14], and in more detail also in Ref. [18], we saw how and why the form (18) was sufficiently enhanced at intermediate Q2 to lead to successful phenomenology when used in quark gap SD equation (5) and bound-state BS equation (3) through Eq. (1), at least in the case when the contribution of the dimension-4 (F2) condensate to CA and CG in Eq. (12), was given by Eq. (13). Acknowledgment The authors gratefully acknowledge the support of the Croatian Ministry of Science and Technology contracts 0119261 and 0098011. References 1. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147,385 (1979); ibid. 448 (1979). 2. L. S. Celenza and C. M. Shakin, Phys. Rev. D 34,1591 (1986). 3. M. J. Lavelle and M. Schaden, Phys. Lett. B 208, 297 (1988). 4. M. Lavelle and M. Schaden, Phys. Lett. B 246, 487 (1990). 5. J. Ahlbach, M. Lavelle, M. Schaden and A. Streibl, Phys. Lett. B 275,124 (1992). 6. M. Lavelle and M. Oleszczuk, Mod. Phys. Lett. A 7, 3617 (1992). 7. P. Boucaud, A. Le Yaouanc, J. P. Leroy, J. Micheli, O. Pene and J. Rodriguez-Quintero, Phys. Lett. B 493, 315 (2000). 8. F. V. Gubarev, L. Stodolsky and V. I. Zakharov, Phys. Rev. Lett. 86, 2220 (2001). 9. F. V. Gubarev and V. I. Zakharov, Phys. Lett. B 501, 28 (2001). 10. K. I. Kondo, Phys. Lett. B 514, 335 (2001). 11. K. I. Kondo, T. Murakami, T. Shinohara and T. Imai, Phys. Rev. D 65, 085034 (2002). 12. P. Boucaud ei al, JHEP 0304, 005 (2003). 13. D. Dudal and H. Verschelde, J. Phys. A 36, 8507 (2003). 14. D. Kekez and D. Klabucar, arXiv:hep-ph/0307110. 15. A. A. Slavnov, arXiv:hep-th/0407194. 16. E. Ruiz Arriola, P. O. Bowman and W. Broniowski, arXiv:hep-ph/0408309. 17. X. d. Li and C. M. Shakin, arXiv:nucl-th/0409042. 18. D. Klabucar and D. Kekez, "(A2)-condensate and Dyson-Schwinger approach to mesons", proceedings of Bled Workshops in Physics, Vol. 4, No. 1, pp. 47-56, edited by B. Golli, M. Rosina and S. Sirca; published by DMFA - zaloznistvo, Ljubljana 2003, ISSN 1580-4992. 19. R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001). 20. P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003). 21. P. Jain and H. J. Munczek, Phys. Rev. D 48, 5403 (1993). 22. P. Maris and C. D. Roberts, Phys. Rev. C 56, 3369 (1997). 23. P. Maris and P. C. Tandy, Phys. Rev. C 60, 055214 (1999). 24. C. D. Roberts, arXiv:nucl-th/0304050. 25. R. Alkofer, C. S. Fischer and L. von Smekal, Acta Phys. Slov. 52,191 (2002). 26. C. S. Fischer and R. Alkofer, Phys. Rev. D 67, 094020 (2003). 27. R. Alkofer, C. S. Fischer and L. von Smekal, Prog. Part. Nucl. Phys. 50, 317 (2003). 28. J. C. R. Bloch, Few Body Syst. 33,111 (2003) [arXiv:hep-ph/0303125]. 29. M. Lavelle, Phys. Rev. D 44, 26 (1991). 30. B. L. Ioffe and K. N. Zyablyuk, Eur. Phys. J. C 27, 229 (2003), and references therein. Vertex Interactions and Applications W. H. Klink Department of Physics and Astronomy University of Iowa, Iowa City, Iowa, USA Abstract. A relativistic quantum mechanics is formulated in terms of four-momentum operators. The free part of the four-momentum operator is built from irreducible representations of the Poincare group, while the interacting part comes from integrating a vertex operator over the forward hyperboloid. If the Fock space on which these operators act is truncated, the Poincare commutation relations no longer hold. But a relativistic few-body theory can still be formulated by using the vertex to define an interacting mass operator. Applications of these ideas are also briefly discussed. 1 Formulation of Relativistic Quantum Mechanics in terms of Vertex Interactions The foundations of nonrelativistic quantum mechanics can be formulated in terms of representations of the Galilei group, the ten parameter group of transformations connecting different inertial frames. Irreducible representations of the Galilei group provide the Hilbert space for free particles, while the unitary operators representing elements of the group specify the connection between wavefunc-tions in different inertial frames [1]. The exponential of these unitary operators also generate operators such as the momentum, angular momentum (including spin), position and free energy operators. For example, if ^(p) is the momentum space wavefunction for a spinless particle of mass m, and Uv = e-imvX is the unitary operator representing a boost, a transformation from one inertial frame to another given by x —> x + vt, then the wavefunction in the boosted frame is given by (Uv^)(p) = e-imvX^(p) = ^(p + mv), where X is the position operator, which is in a momentum representation. Few-body quantum mechanics can then be formulated on tensor products of irreducible representation spaces. However, it is simpler to formulate a many-body theory by introducing creation and annihilation operators with the same arguments as one-particle states, and which transform under Galilei transformations as one particle states. Then the free Hamiltonian and momentum operators can be written as P2 H(fr)=^ d3p|-at(pms)a(pms) (1) P(fr) = X d3 PPat (Pms )a(pms) (2) and must satisfy a number of commutation relations in order to have a Galilei covariant theory. Interactions are introduced by modifying the free Hamiltonian in such a way that the commutation relations are preserved. For example an n-body kernel must be rotationally and Galilei boost invariant in order to satisfy the commutation relations. The procedure just outlined can be generalized to relativistic systems. Rel-ativistic transformations are generated by Poincare transformations, in which a space-time point x is transformed to x = Ax + a, where A is a Lorentz transformation, and a a space-time four-vector translation. Irreducible representation spaces for particles of mass m and spin j are most simply realized as functions over the forward hyperboloid specified by the four-velocity satisfying v ■ v = 1, which is related to the four-momentum by p = mv. Then the Hilbert space for a particle of mass m and spin j is H = L2 (v) x Vj, where Vj is the usual 2j + 1 dimensional spin space[2]. From a relativistic state |v, ct), where ct is the relativistic spin projection ranging between —j and j, a many-particle Fock space is generated by creation and annihilation operators satisfying [a(v, ct), at (v , ct )]± = v053(v — v )5CTCT'. To satisfy locality requirements needed for the vertex interactions, it is necessary to also introduce antiparticle creation and annihilation operators satisfying the same commutation relations. Then the free four-momentum operator can be written as P|_l(fr)=mV -VufQtfv^Jatv.tr) v0 +bt (v,CT)b(v,CT));(3) [P^ (fr),Pv(fr)]= 0 (4) V(" )UA = (A ) Hr v UA P^frjUX1 =(A-1); Pv(fr), (5) where the last two equations guarantee the commutation relations of the Poincaré group. Interactions are generated by vertices with the following properties, the first of which is a locality requirement: [V(x), V(y)] = 0, (x -y)2 < 0, (6) UQV (x)U-1 = V (x + a), (7) UAV (x)UX1 = V (Ax);(8) P^ (I): d4x5(x ■ x - 1)6(x0 )x^V(x), (9) where the interacting four-momentum operator in Eq.(9) is obtained by integrating the vertex operator over the forward hyperboloid. It then follows that the interacting four-momentum operator satisfies [P^(I),Pv (I)]= 0, (10) Ua P^(I)UA1 =(A-1); Pv(I), (11) so that the Poincare conditions are satisfied for the interacting four-momentum operator[3]. The total four-momentum operator is the sum of free and interacting four-momentum operators and satisfies [P^Pv] = [P^(fr) + P^(I),Pv(fr) + Pv(I)] = [P^fr),Pv(I)]-[Pv(fr),PH)] d4x5(x ■ x — 1) 0 0 = 0. Vertex operators satisfying the above properties are generally made out of local fields. For the local charged scalar field ®(x) d3v -(e-ip'xa(v) + eip'xbt(v))> (12) vo an example of a vertex operator is V(x) = lanDt1'-where D^...^ = and the an are constant coefficients. The local charged scalar field can readily be generalized to include spin [4] and internal symmetries, including gauge transformations. 2 Application of Vertex Interaction to a Charged Particle in an External Electromagnetic Field Let P(x) be the (local) electromagnetic currrent operator for a particle of mass m and spin j. Such an operator can be written as a linear combination of bilinears in creation and annihilation operators and hence is an element of a representation of U(N,N) on Fock space[4]. A vertex can be defined by V(x) = P(x)A®xt(x) and the four-momentum operator for a particle in an external electromagnetic field is given by P|i = P^(fr) + P^(ext), Pp.(e*t) = d4xö(x ■ x — 1)x^6X0 V(x);(13) ¥t = e-1P0t¥ (14) gives the time evolution of the system. But the exponential of the energy operator is an element of the representation of U(N,N) on Fock space and the action of this representation is known[5]. Hence one can use such a vertex interaction to explicitly calculate particle production from a time independent external electromagnetic field. 3 Bakamjian-Thomas Mass Operators in terms of Vertex Interactions If the Fock space on which the vertex generated four-momentum operators act is truncated, the components of the four-momentum operator will no longer com- mute. A procedure for constructing a commuting four-momentum operator is to use the vertex operator to define an interacting mass operator on the truncated Fock space. To prepare for the construction of interacting mass operators, introduce velocity states, states with simple Lorentz transformation properties. If a Lorentz transformation is applied to a many-particle state, |pi, ci ...pn, on.) = a (pi, oi)... at (pn, on)|0), then it is not possible to couple all the momenta and spins together to form spin or orbital angular momentum states, because the Wigner rotations for each momentum state are different. However, velocity states, defined as n-particle states in their overall rest frame boosted to a four-velocity v will have the desired Lorentz transformation properties: |v, ki, m) : = UB(v)|ki , m ...kn, Mn) (15) = 2>1,Ol ...pn,On)n (RWi ). (16) UA|v,ki, m) = UAUB(v)|ki > M-i ...kn> Mn ) = UB(Av)UR |ki > M-i ...kn> Mn) = Y |Av,Rwki,Mi) n D'm' a (Rw). (17) Now all the Wigner rotations in the D functions are the same. Moreover the same Wigner rotation also multiplies the internal momentum vectors, which means that for velocity states, spin and orbital angular momentum can be coupled together exactly as is done nonrelativistically. The relationship between single particle and internal momenta is given by pi = B (v)k,Y_ ki = 0; From the definition of velocity states it then follows that V|>,ki, Mi) = vjv,ki, Mi), (18) M(fr)|v,ki, Mi) = mf|v,ki,m), (19) P^(fr)|v, ki ) = mf^|v, ki,m), (20) with mf = Y. \/mi + kt2 the free 'mass' of the n-particle velocity state and P|i(fr) = M(fr)V^. On velocity states the free four-momentum operator has been written as the product of the four-velocity operator times the free mass oper-ator[6]. Four-momentum operators are written as P^ = MV^, where the four-velocity operator is defined by V^ := ^(frj • ^he mass operator is the sum of free and interacting mass operators, M = M(fr) + M(I); if the mass operator commutes with the four-velocity operator and Lorentz transformations, then the Poincare commutation relations, Eqns.(4),(5), are satisfied. Since the four-velocity of the overall system is kinematic, it can be ignored; what remains then is to solve the mass operator eigenvalue equation, M¥ = m¥. Now the vertex operator at the space-time point 0 is a Lorentz scalar. Velocity state matrix elements of V(0) can then be used to generate an interacting mass operator: M(I) := + (Kt(--)K-Mhy)|¥3Q) =m|¥3Q) (26) m — M3Q-M In addition to MGBE, Eq.(26) has an additional mass operator which gives the coupled channel contribution. With this additional term it should be possible to use perturbation theory to compute widths and level shifts. These and other applications [9] show the utility of a coupled channel approach to few-body systems, in which the interactions are generated by vertices arising from quantum field theory. References 1. J. Levy-Leblond, Jour. Math. Phys. 4 (1963) 776; W. Klink, R. Warren, Jour. Math. Phys. 11 (1970) 1155. 2. W. H. Klink, Ann. Phys. 213 (1992) 31. 3. W. H. Klink, Phys. Rev. C58 (1998) 3587. 4. W. H. Klink, Nucl. Phys. A 716 (2003) 136. 5. Mark G. Davidson, Pac. J. Math. 129 (1987) 33. 6. W. H. Klink, Nucl. Phys. A 716 (2003) 158. 7. A. Krassnigg, W. Schweiger, W. Klink, Phys. Rev. C67 (2003)064003. 8. L. Ya. Glozman, W. Plessas, K. Vargas, and R. F. Wagenbrunn, Phys. Rev. C57 (1998) 3406; C58 (094030) 1998. 9. M. Lechner, W. Schweiger, and W. Klink, Proceedings of the 17th IUPAP Conference on Few-Body Problems in Physics, W. Gloeckle, W. Tornow editors, Elsevier, Amsterdam (2004) pg. S258, gives a preliminary account of using vertex interactions to compute positronium bound states. Strong Decays of Baryons* T. Meldea, L. Cantonb, W. Plessasa, and R. F. Wagenbrunna a Theoretical Physics, Institute for Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria b INFN Sezione di Padova and Dipartimento di Fisica, Via Marzolo 8, Padova, Italy Abstract. A Poincare-invariant description of mesonic baryon resonance decays is presented following the point form of relativistic quantum mechanics. In this contribution we focus on pionic decay modes. It is found that the theoretical results in general underestimate the experimental ones considerably. Furthermore, the problem of a nontrivial normalization factor appearing in the definition of the decay operator is investigated. The present results for decay widths suggest a normalization factor that is consistent with the choice adopted for the current operator in the studies of electroweak nucleon form factors. 1 Introduction Constituent quark models (CQMs) provide an effective tool to describe the essential hadronic properties of low-energy quantum chromodynamics. Recently, in addition to the traditional CQM, whose hyperfine interaction derives from one-gluon exchange (OGE) [1], alternative types of CQMs have been suggested such as the ones based on instanton-induced (II) forces [2,3] or Goldstone-boson-exchange (GBE) dynamics [4]. In particular, the GBE CQM aims to include the effective degrees of freedom of low-energy QCD, as they are suggested by the spontaneous breaking of chiral symmetry (SBxS). Over the years, a number of valuable insights in strong decays of baryon resonances have been gained by various groups, e.g., in refs. [5-9]. Nonetheless, one has still not yet arrived at a satisfactory explanation especially of the N and A resonance decays. This situation is rather disappointing from the theoretical side, especially in view of the large amount of experimental data accumulated over the past years [10]. Here, we study the mesonic decays of baryon resonances along relativistic, i.e. Poincare-invariant, quantum mechanics [11]. This approach is a-priori distinct from a field-theoretic treatment. It assumes a finite number of degrees of freedom (particles) and relies on a relativistically invariant mass operator with the interactions included according to the Bakamjian-Thomas construction [12] thereby fulfilling all the required symmetries of special relativity. We assume a decay operator in the point-form spectator approximation (PFSA) with a pseudovector coupling. The PFSA has already been applied to the calculation of electromagnetic and axial form factors of the nucleons [13-15] and electric radii as well as * Talk delivered by T. Melde magnetic moments of all octet and decuplet baryon ground states [16]. In all cases the experimental data are described suprisingly well within this approach. Covariant results for the strong decays of N and A resonances have already been presented in ref. [17] for the relativistic GBE and OGE CQMs. They show a dramatically different behaviour as compared to previous non-relativistic calculations [18,19]. Specifically, it turns out that the theoretical results, in general, underestimate the experimental ones considerably. This behaviour has also been observed in the relativistic calculation based on the Bethe-Salpeter equation using instanton-induced dynamics [20]. Up till now all relativistic approaches face the problem of defining appropriate decay operators. Usually one has resorted to simplified versions such as the spectator model. 2 Theory Generally, the decay width r of a resonance is defined by the expression r = 2npf |F (wf)|2 , (1) where F (i —> f) is the transition amplitude and pf is the phase-space factor. In eq. (1) one has to average over the initial and to sum over the final spin-isospin projections. Previous calculations, based on nonrelativistic approximations of the transition amplitude encountered an ambiguity in the proper definition of the phase-space factor [7,21,22]. Here, we present a Poincare-invariant definition of the transition amplitude, thereby resolving this ambiguity. In particular, we adhere to the point-form of relativistic quantum mechanics [11], because in this case the generators of the Lorentz transformations remain purely kinematic and the theory is manifestly covariant [23]. The interactions are introduced into the (invariant) mass operator following the Bakamjian-Thomas construction [12]. The transition amplitude for the decays is defined in a covariant manner, under over-aU momentum conservation (p^ — = Qn F (i->f) = stem from the velocity-state representations of the baryon states 0 , °2, °3) I M3M'3 , , 5 , . ., = \ —-rrr---r3rgqq7tu (p!, ) y y A u (p,, cr1) V (L ^i) (L ^i) 2p2°5 (P2 — P2') 2p30 5 (P3 — P3') Qn^ (4) The overall momentum conservation, P^—P^ = together with the two spectator conditions define the relation between all incoming and outgoing quark momenta. In particular, the momenta of the active quark are related by pi — p/ = Q, where Q is completely determined. Thus the momentum transferred to the active quark is different from the momentum transfer to the baryon as a whole. This is a consequence of translational invariance which thereby introduces effective many-body contributions into the above definition of the spectator-model decay operator. Furthermore, in eq. (4) there appears an overall normalization factor ^ M3 M'3 3. (5) V (L ^i) (L ^i) Through the and the on-mass-shell condition of the quarks it depends on the individual quark momenta. This choice of N is consistent with the one used in the definition of the electromagnetic and axial currents in the PFSA calculations of the nucleon electroweak form factors by the Graz-Pavia collaboration [13-15]. It guarantees for the correct proton charge. However, it is not a unique choice. Any other normalization factor of the asymmetric form would do the same. In order to study the effects of these further choices we investigate the dependence of the decay widths on the parameter range 0 < y < 1. 3 Results The decay widths of the PFSA calculation with the decay operator given in eq. (4), with the symmetric normalization factor, are shown in table 1 for the GBE and OGE CQMs. It is immediately seen that only the N|535 and N|7i0 predictions are Table 1. PFSA predictions for n decay widths of the relativistic GBE [4] and OGE [9] CQMs in comparison to the Bethe-Salpeter results of the II CQM [20] and experimental data [24]. In the last three columns the theoretical results are expressed as percentage fractions of the (best-estimate) experimental values in order to be compared to the measured An branching ratios. Decays Experiment Rel. CQM An % of Exp. Width -> Nn GBE OGE II branching ratio GBE OGE II N*440 (227 ± 18)-59 33 53 38 20 - 30% 14 24 17 N*520 (66 ± 6)+ 5 17 16 38 15 - 25% 26 24 58 N *535 (67 ± 15)-27 90 119 33 < 1% 134 178 49 N *650 (109 ± 26)-36 29 41 3 1 - 7% 27 38 3 N1675 (68 ± 8)-14 5.4 6.6 4 50 - 60% 8 10 6 N1700 (10 ±5)- 3 0.8 1.2 0.1 > 50% 8 12 1 ^710 (15 ±5)-30 5.5 7.7 n/a 15 - 40% 37 51 n/a Al232 (119 ± 1)- 5 37 32 62 - 31 27 52 Al600 (61 ± 26)-20 0.07 1.8 n/a 40 - 70% « 0 3 n/a Al620 (38 ± 8)- 6 11 15 4 30 - 60% 29 39 11 A1700 (45 ± 15)-20 2.3 2.3 2 30 - 60% 5 5 4 within the experimental range. All other decay widths are underestimated, some of them considerably. In this regard, it is noteworthy that in the case of the N|535 the An branching ratio is exceptionally small (< 1 %). Therefore we found it interesting to look at the results with a view to the measured An branching ratios. In fact, one can observe a striking relation between these branching ratios and the sizes of the theoretical decay widths, expressed as percentage fractions of the experimental values in the last three columns of table 1: The larger the An branching ratio of a resonance, the bigger the underestimation of the (best-estimate) experimental value. This observation hints to a possible systematic problem in the description of mesonic decay widths within (relativistic) CQMs. It calls for a more complete treatment of baryon resonances with a more realistic coupling to decay channels. In fig. 1 we demonstrate the dependence of the PFSA predictions (for the case of the GBE CQM) on the possible asymmetric choice of the normalization factor N (see eq. (6)). In the range 0 < y < 1 all decay widths grow rapidly with increasing y. In this way one could enhance the theoretical predictions considerably. However, if one wants neither one of the decay widths to exceed its experimental range, one is limited to a value of y < 0.5. Any y lower than 0.5 would lead to decay widths much too small in most cases. Consequently, a symmetric normalization factor as in eq. (4) seems to be the preferred and most reasonable choice also in the context of hadronic decay widths. Nucleon Decay Widths Symmetry Factor y Delta Decay Widths Fig. 1. Dependence of some resonance decay widths on the choice of the normalization factor after eq. (6). 4 Summary We have presented a Poincare-invariant description of strong baryon resonance decays in point form within relativistic CQMs. Covariant predictions have been given for n decay widths. They are considerably different from previous nonrel-ativistic results or results with relativistic corrections included. The covariant results calculated with a spectator-model decay operator show a uniform trend. In almost all cases the corresponding theoretical predictions underestimate the ex- perimental data considerably. This is true in the framework of Poincaré-invariant quantum mechanics (here in point form) as well as in the Bethe-Salpeter approach [20]. Indications have been given that for a particular resonance the size of the underestimation is related to the magnitude of the An branching ratio. This hints to a systematic defect in the description of the decay widths. The investigation of different possible choices for a normalization factor in the spectator-model decay operator has led to the suggestion that the symmetric choice is the most natural one. It is also consistent with the same (symmetric) choice that had been adopted before for the spectator-model current in the study of the electroweak nucleon form factors. This work was supported by the Austrian Science Fund (Project P16945). T.M. would like to thank the INFN and the Physics Department of the University of Padova for their hospitality, and MIUR-PRIN for financial support References 1. S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986). 2. U. Loering, K. Kretzschmar, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A10,309 (2001). 3. U. Loering, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A10, 395 (2001). 4. L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998). 5. F. Stancu and P. Stassart, Phys. Rev. D 39, 343 (1989). 6. S. Capstick and W. Roberts, Phys. Rev. D 47,1994 (1993). 7. P. Geiger and E. S. Swanson, Phys. Rev. D 50, 6855 (1994). 8. E. S. Ackleh, T. Barnes, and E. S. Swanson, Phys. Rev. D 54, 6811 (1996). 9. L. Theussl, R. F. Wagenbrunn, B. Desplanques, and W. Plessas, Eur. Phys. J. A12, 91 (2001). 10. S. A. Dytman, and E. S. Swanson (Eds.): NSTAR 2002 (Proc. of the Workshop on the Physics of Excited Nucleons, Pittsburgh, Pennsylvania, 2002). Singapore: World Scientific 2003. 11. B. D. Keister and W. N. Polyzou, Adv. Nucl. Phys. 20, 225 (1991). 12. B. Bakamjian and L. Thomas, Phys. Rev. 92,1300 (1953). 13. R. F. Wagenbrunn et al., Phys. Lett. B511, 33 (2001). 14. L. Y. Glozman et al., Phys. Lett. B516,183 (2001). 15. S. Boffi et al., Eur. Phys. J. A14,17 (2002). 16. K. Berger, R. F. Wagenbrunn, and W. Plessas, nucl-th/0407009 (2004). 17. T. Melde, W. Plessas, and R. F. Wagenbrunn, Few-Body Syst. Suppl. 14, 37 (2003). 18. A. Krassnigg et al., Few Body Syst. Suppl. 10, 391 (1999). 19. W. Plessas et al., Few Body Syst. Suppl. 11, 29 (1999). 20. B. Metsch, hep-ph/0403118 (2004). 21. S. Kumano and V. R. Pandharipande, Phys. Rev. D 38, 146 (1988). 22. R. Kokoski and N. Isgur, Phys. Rev. D 35, 907 (1987). 23. W. H. Klink, Phys. Rev. C 58, 3587 (1998). 24. S. Eidelman et al., Phys. Lett. B592,1 (2004). Bled Workshops in Physics Vol. 5, No. 1 Relativistic Treatment of Baryon Reactions W. Plessas Theoretical Physics, Institute for Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria Abstract. We give a survey of the performance of modern relativistic constituent quark models in the description of baryon properties and reactions. In particular, we address baryon spectroscopy, elastic electromagnetic and axial nucleon form factors, electric radii and magnetic moments of the octet and decuplet baryon ground states, electromagnetic transitions, as well as mesonic baryon resonance decays. Directions for further improvements of constituent quark models are indicated. 1 Relativistic Constituent Quark Models Constituent quark models (CQMs) represent a powerful tool in modern hadronic physics. They serve as an effective description of hadron properties at low and intermediate energies. CQMs have undergone a vivid development over the past few years. Notably, one has found that CQMs must take into account the relevant properties of quantum chromodynamics (QCD) in the nonperturbative regime and have to fulfill the requirements of a relativistic theory. In order to arrive at a reasonable description of hadron phenomena, CQMs should meet the symmetry requirements of both (low-energy) QCD and special relativity. From the outset, CQMs rely on a finite number of degrees of freedom. One assumes a few-quark system, {QQ} or {QQQ} etc., with certain internal interactions and solves the corresponding dynamical equations. The theory should be covariant. Thus, it is most natural to resort to a Poincare-invariant relativistic quantum theory. Such an approach is well defined and it can be solved rigorously, at least for confined two- and three-quark systems. In particular, one solves the eigenvalue problem of the invariant mass operator for a given CQM, obtains the eigenenergies and eigenstates, and can go ahead to calculate reactions involving the corresponding hadron states. If one uses relativistic operators and carries out the necessary Lorentz transformations exactly, one will arrive at covariant predictions for the observables in question. The latter task is most efficiently achieved in the point-form version of relativistic quantum mechanics (RQM), since in this case the generators of Lorentz boosts remain purely kinematical. It should be noted that for a CQM to be considered as 'relativistic' it is not necessary that the inherent dynamics is derived in a relativistic manner, e.g., from a quantum field theory. It suffices that the mutual interactions between the constituents of a given system meet the requirements of Poincare invariance. One could even introduce phenomenologically motivated interactions into the Hamil-tonian (or equivalently into the mass operator). Once such an interacting Hamil-tonian fulfills the commutator relations of the Poincare algebra, all the symmetries of a Lorentz-covariant theory can be implemented. Recently, we have seen interesting new results for baryons especially from the relativistic CQMs that rely on one-gluon-exchange (OGE), Goldstone-boson-exchange (GBE), and instanton-induced (II) effective interactions between confined constituent quarks. In this paper, we concentrate on the GBE CQM by the Graz group [1], a relativistic version [2] of the Bhaduri-Cohler-Nogami (BCN) OGE CQM [3], and the II CQM by the Bonn group [4]. The first two are constructed in the framework of RQM, while the last one is formulated in the Bethe-Salpeter approach. Incidentally, the main differences lie in the hyperfine interactions, while the (linear) confinement potential is very similar in all cases; its strength is practically compatible with the string tension of QCD. As a result, the local extensions of the corresponding {QQQ} states are also commensurable. They are much narrower than in nonrelativistic CQMs, which use a confinement potential with an unreasonably weak strength. 2 Baryon Spectroscopy The detailed light and strange baryon spectra of the GBE, OGE, and II CQMs can be found in the original papers [1,9,2]. A critical discussion of the qualitative differences between the GBE and OGE hyperfine interactions is presented in ref. [5]. For a critique of some erroneous and misleading results in the literature see also ref. [6]. In the comparison of the GBE, OGE, and II CQMs some relevant observations are to be made specifically with regard to the N and A spectra as exemplified in Figure 1. Only the GBE CQM can provide for the correct level orderings of positive-and negative-parity excitations in the N spectrum. The \ N(1440) Roper resonance cannot be brought down below the \ N (1535) resonance by the OGE and II CQMs (as long as the correct N-A splitting is maintained). The success of the GBE CQM is due to its particular spin-flavor dependence in the hyperfine interaction. It is also favourable for reproducing simultaneously the right level ordering in the A spectrum (which is opposite to the nucleon case). However, all CQMs fail to describe the lowest excitation in the A spectrum, the \ A( 1405) resonance, at the right energy. Most probably this is due to the limitation to {QQQ} configurations only. Here, an intriguing shortcoming of the present CQMs becomes evident with respect to a realistic description of baryon resonances, namely, the missing coupling to decay channels. The same conclusion can be drawn from studies of inelastic electromagnetic reactions and mesonic resonance decays (cf. also the discussion below). 3 Elastic Electroweak Nucleon and Baryon Structure An immediate application of any CQM for baryons is the calculation of elastic electromagnetic and axial nucleon form factors. It provides a stringent test of the quality of the nucleón wave functions. All of the relativistic CQMs considered here have been studied in this respect. A (partial) comparison is presented in Figure 2. The covariant predictions of the GBE CQM for electromagnetic and weak nucleon form factors calculated in point-form spectator approximation (PFSA) are published in refs. [8,14,15]. The electric radii and magnetic moments of the octet and decuplet ground states are presented in ref. [11]. In all cases a remarkable good agreement of the direct predictions of the GBE CQM with the existing experimental data is found. The PFSA calculation is most favourable for reaching a consistent explanation of the electroweak nucleon structure at low momentum transfers. The good quality of the results, being covariant and practically current-conserving, is not yet fully understood in detail. Large differences of the theoretical predictions are found with the nonrelativistic impulse approximation (NRIA) [8,14-16]; see also Figure 3 below. It has become evident that a nonrelativistic theory does not work for the nucleon form factors. This is even true with regard to the electric radii and magnetic moments, i.e. with observables in the limit of zero momentum transfer. The comparison of the GBE and OGE CQMs, both calculated in PFSA, tells us that there is no big influence from the type of the nucleon wave function [12]. As soon as the nucleon wave function is realistic, especially with the right spatial extension and the correct (mixed symmetry) spatial components contained, the nucleon form factors will be predicted quite reasonably if the relativistic effects are properly taken into account. At least the influences of different dynamics in the CQM are much smaller than relativistic effects. Only if an oversimplified wave function is employed, such as a completely symmetric SU(6) one, like in the case of confinement only, the description evidently falls short (cf., e.g., the corresponding results for the neutron electric form factor shown in Figure 2). In general, the PFSA predictions are also rather similar to the results obtained for M [MeV] 180017001600150014001300120011001000- M [MeV] IfiOO- 1800 -1700 ItiOO-150014001300120011001000- Fig. 1. Nucleon (left) and A (right) excitation spectra of three different types of relativistic CQMs. In each column the left horizontal line represents the results of the BCN OGE CQM, as parametrised in [2], the middle one of the II CQM (version A) [4], and the right one of the GBE CQM [1]. The shadowed boxes give the experimental data with their uncertainties after the latest compilation of the PDG [7]. 1.0 1.........' ' .......111 .......1 ' 1 1 O Andivahis V Walker Gp □ Sill e A Hoehler o Bartel - GBE --- OGE ---- Conf. A II 0.5 ■ 'vL \\ \ % - 1.........1 Andivahis V Walker □ Sill A Hoehler O Bartel - GBE ---OGE ---- Conf. ...... II Q2 [(GeV/c)2] Q2 [(GeV/c)2] 0 2 3 4 0 2 3 4 x Milbrath + Jones * Gayou O Andivahis V Walker A Hoehler --GBE ---OGE ---Conf. ..........II □ Pion world data O Pion Mainz A Neutrino world data --GBE ---OGE ---Conf. II 0 1 2 3 Q2 [(GeV/c)2] Q2 [(GeV/c)2] 4 5 Fig. 2. Predictions of different CQMs for the nucleon electromagnetic and axial form factors. The solid and dashed lines represent the PFSA results for the GBE CQM [1] and the BCN OGE CQM [2], respectively; the dash-dotted lines refer to the case with confinement only. The dotted lines show the results of the II CQM [4] within the Bethe-Salpeter approach after ref. [13]. the II CQM following the Bethe-Salpeter approach [13]. This observation is remarkable in view of the differences in the dynamics of the CQMs and the distinct frameworks of the calculations. One may interpret these findings in such a way that for the nucleon ground states any degrees of freedom other than {QQQ} are presumably unimportant (at least in the low-momentum-transfer range considered here) and the relativistic current in the spectator approximation is working quite well. If the relativistic boost effects are properly included in the calculation of the matrix elements of elastic form factors and covariant results are thus obtained, a rather consistent explanation of all experimental data becomes possible. The boost effects are taken into account accurately in the point-form calculations; the same is claimed for the Bethe-Salpeter approach [13]. In order to elucidate the peculiarities of the PFSA further, the Graz group has recently undertaken analogous calculations of the nucleon electromagnetic form factors in instant-form spectator approximation (IFSA). The comparison is given in Figure 3 for the case of the GBE CQM. It is seen that the IFSA predictions obtained with the same wave functions as in the case of the PFSA, without introducing any additional parameters, can by no means explain the experimental data. In some way the IFSA results even resemble the ones from the NRIA. In addition to this obvious shortcomings in the comparison with phenomenology, the IFSA must be rejected because it does not account for the correct boost effects and it is not covariant (frame-independent). In instant form the generators of the boosts are a-priori interaction-dependent. Instead, in all IFSA calculations so far, like in the one of Figure 3, free boost transformations have been employed. We consider this as a notorious problem of approximative instant-form approaches. The IFSA results shown in Figure 3 were calculated in the Breit frame. They would be different in another reference system, such as the laboratory frame. Clearly, one cannot rely on such results. Contrary to that the PFSA is manifestly covariant and the corresponding predictions are frame-independent. The point-form approach works remarkably well in the case of elastic nucleon form factors; the same is true with regard to electric radii and magnetic moments of all octet and decuplet baryon ground states (as far as we can compare to existing experiments) [11]. Still, one has to bear in mind that the PFSA calculation is approximative and incomplete. In particular, explicit many-body currents are still missing. On the other hand, one meets Lorentz covariance and seemingly also the continuity equation is fulfilled to a good extent. The latter property has been tested by calculating the matrix element of the third component of the current operator j3 (q). This matrix element must vanish exactly in the Breit frame if the current is conserved. Indeed, the numerical values we obtain are extremely small. In the range of momentum transfers considered here, the magnitude of the matrix element of j3 (q) remains lesser than 1 % of the zeroth component j0 (q) (from which the electric form factor is deduced). 4 Electromagnetic Transition Form Factors The next step is testing the relativistic CQMs in yN —> N* reactions. First results in this regard have been gained recently for the GBE CQM in PFSA [14]. In Fig. 3. Comparison of proton and neutron electromagnetic form factors of the GBE CQM [1] calculated in PFSA and IFSA as well as in NRIA. Figure 4 covariant predictions for helicity amplitudes of y-induced transitions to several N * resonances are shown. They were calculated in a manner completely analogous to the elastic nucleon form factors. Data are still scarce and have relatively large uncertainties. In case of the neutron there are only data at Q 2 = 0. The theoretical predictions appear reasonable even though one finds deviations from the experimental data that are bigger than in the elastic case. We do not yet know the definite reasons for the discrepancies. One may suspect that the description of the resonances in a CQM with {QQQ} configurations only is not realistic enough, as soon as resonances are involved. Further investigations in this field are urgently needed. 5 Mesonic Decays of Baryon Resonances Another wide field for applications of CQMs are the decays of baryon resonances. Preliminary relativistic predictions for widths of pionic decays of N* and A* resonances were already presented in ref. [16]. In that work one produced first covari- -100 -120 -140 120 100 80 60 _> 40 a — 20 " 0 — proton — neutron N(1520) ..... Iii 2 Q2 [(GeV/c)2] i.......... - * N(1520) r —--i - ______—--------1----- : — proton - — neutron . .........,.........,.........,.........- 2 Q2 [(GeV/c)2] N(1650) — proton — neutron proton neutron N(1675) Q2 [(GeV/c)2] 1.........1.........1.........1......... t — proton _ — neutron . r....." N(1675) : r: 2 Q2 [(GeV/c)2] — proton — neutron 2 Q2 [(GeV/c)2] N(1700) ......3 ' Fig. 4. PFSA predictions of the GBE CQM [1] for helicity amplitudes of yN N * transitions. The data at Q2 = 0 are from the PDG [7] for the proton (solid circles) and the neutron (open circles). The proton data at Q2 >0 are taken from the review [15] and references cited therein. 150 100 50 0 4 4 0 3 60 40 C5 20 0 0 2 3 4 2 2 „„.,„..2 4 Q [(GeV/cri Q [(GeV/cn 20 20 0 0 2 3 4 4 Q [(GeV/cri 40 20 0 0 2 3 4 4 ant results with the GBE and OGE CQMs using a spectator model decay operator in point form (PFSA). The relativistic PFSA results were found to be quite distinct from previous results of nonrelativistic calculations or calculations with relativis-tic corrections. Almost all decay widths turned out to be (much) too small. Only in two cases, namely in N(1535) and N(1710), the magnitude of the experimental value for the n-decay width is reached. In the meantime the calculations have been improved and further investigations have been undertaken, such as the calculation of n-decay modes [17]. Also the Bonn group has calculated decay widths with the II CQM in the Bethe-Salpeter approach [18]. They found results that are again quite similar to the ones obtained for the GBE CQM in PFSA. In particular, they confirmed the general trend of the decay widths resulting too small. Obviously an important ingredient is missing in the present description. Once more we are hinted to the necessity of taking into account the coupling to the decay channels and including explicit mesonic degrees of freedom. For a further and more detailed discussion of these aspects see T. Melde's contribution to this Workshop [19]. Acknowledgment The results discussed in this paper rely on essential contributions by my colleagues K. Berger, L. Glozman, T. Melde, and especially R. Wagenbrunn (Graz) as well as on collaborations with W. Klink (Iowa), S. Boffi, and M. Radici (Pavia). This work was supported by the Austrian Science Fund (Project P16945). References 1. L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998). 2. L. Theussl, R. F. Wagenbrunn, B. Desplanques, and W. Plessas, Eur. Phys. J. A12, 91 (2001). 3. R. K. Bhaduri, L. E. Cohler, and Y. Nogami, Nuovo Cim. A65, 376 (1981). 4. U. Loering, K. Kretzschmar, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A10,309 (2001); ibid., 395 (2001). 5. L. Ya. Glozman, Z. Papp, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. C 57, 3406 (1998). 6. W. Plessas, Few-Body Syst. Suppl. 15,139 (2003). 7. S. Eidelman et al., Phys. Lett. B592,1 (2004). 8. R. F. Wagenbrunn, S. Boffi, W. Klink, W. Plessas, and M. Radici, Phys. Lett. B511, 33 (2001). 9. L. Y. Glozman, M. Radici, R.F. Wagenbrunn, S. Boffi, W. Klink, and W. Plessas, Phys. Lett. B516,183 (2001). 10. S. Boffi, L.Y. Glozman, W. Klink, W. Plessas, M. Radici, and R.F. Wagenbrunn, Eur. Phys. J. A14,17 (2002). 11. K. Berger, R. F. Wagenbrunn, and W. Plessas, arXiv:nucl-th/0407009 (2004), to appear in Phys. Rev. D. 12. R. F. Wagenbrunn, S. Boffi, L. Y. Glozman, W. Klink, W. Plessas, and M. Radici, Few Body Syst. Suppl. 14, 411 (2003). 13. D. Merten, U. Loering, K. Kretzschmar, B. Metsch, and H.R. Petry, Eur. Phys. J. A14, 477 (2002). 14. R.F. Wagenbrunn, Contribution to the 19th European Conference on Few-Body Problems, Groningen, 2004, to appear in the proceedings. 15. V. D. Burkert and T. S. H. Lee, arXiv:nucl-ex/0407020, to appear in Int. J. Mod. Phys. E. 16. T. Melde, W. Plessas, and R. F. Wagenbrunn, Few-Body Syst. Suppl. 14, 37 (2003). 17. T. Melde, W. Plessas and R. F. Wagenbrunn, Contribution to the N*2004 Workshop, Grenoble, 2004, to appear in the proceedings; arXiv:hep-ph/0406023. 18. B. Metsch, arXiv:hep-ph/0403118 (2004). 19. T. Melde, L. Canton, W. Plessas, and R. F. Wagenbrunn, Contribution to this Workshop; arXiv:hep-ph/0410274. Bled Workshops in Physics Vol. 5, No. 1 sCX The Double-Charm Hyperons and Their Interactions Dan-Olof Riska Helsinki Institute of Physics and Department of Physical Sciences, 00014 University of Helsinki, Finland Abstract. The main interaction of the ground states of the double-charm hyperons form is due to their light quarks and can be derived from the corresponding nucleon-nucleon interaction by rescaling of the interaction to take into account the difference between interaction strengths for pairs of light flavor quarks and pairs of triplets of light flavor quarks. Nucleons and double-charm (and double-beauty hyperons) are very likely to form bound states in the triplet state. 1 The double-charm hyperons Several double-charm hyperons, the lowest one of which is the ground state multiplet with the Ecc at 3.46 GeV have been found[1]. The corresponding (probably) spin 3/2 multiplet has been found at 3.52 GeV. The ground state multiplet E+c, E++ of the double-charm hyperons forms a spin 1/2 isospin 1/2 multiplet, with the valence quark configuration dcc and ucc [2]. 2 Models for the double-charm hyperons The energy of the ground state multiplet is in the range ~ 3.5 GeV, suggested by early model calculations [2,3]. The Skyrme model slightly underpredicted (40 MeV) the value of the empirically found splitting 60 MeV between the spin 1/2 ground state and the spin 3/2 excited state while the extant lattice value of this splitting is somewhat large value (90 MeV) [4]. 3 Hyperfine splitting structure The discovery of the ground state multiplets of the doubly charmed hyperons allows an overall view of the mass and flavor dependence of the ground state baryon splittings (Table 1). These do not vary smoothly with baryon mass, as heavy quark symmetry would suggest. As an example the splitting in the E spectrum is larger than that in the I spectrum, and the ground state splitting is similar in the single and double-charm hyperons. The splittings may be described by the schematic phenomenological flavor and spin dependent hyperfine interaction model [5]: V: z i = = £ c^|N) . q q q q (2) 60 D.-O. Riska The interaction between two double-charm hyperons that arises from the interaction between the light flavor quarks may be determined from realistic nucleon-nucleon interaction models as eg. the models in refs. [7-9]. From these the corresponding interactions between double-charm hyperons may be derived by application of the appropriate downscaling of the strengths of the corresponding interaction components. With the quark model scaling factors two of the three rescaled nucleon-nucleon interactions models yield that deuteron-like bound states of double-charm hyperons, with binding energies in the range 87 — 457 MeV. In Table 2 contains the calculated binding energies obtained for the deuteron-like states of double-charm hyperons. The difference between these values gives an estimate of the theoretical uncertainty that derives from the different short range behavior of the nucleon-nucleon interaction models. Two-baryon states formed of double-charm hyperons can couple to states with a single charm and a triple-charm Qccc by quark rearrangement. If the latter states have lower energy the former are metastable rather than bound. This depends on the size of the binding energy as compared to the mass difference: Ac = M-ccc+M-cu—2Mccl, where I represents a light quark. For some quark models the inequality Ac <0 holds and for those A « [130 — 158] MeV [10]. Adoption of those values imply that the double charm hyperons form bound states with the AV18 potential, but only metastable states with the Nijm II potential. Double-Charm hyperons Potential Binding Energy (MeV) AV18 -457 (-28) Paris — Nijm II —87 Double-Beauty double-charm hyperons AV18 —603 (—183) Paris —0 Nijm II — 102 Double-Beauty hyperons AV18 —782 (—439) Paris —2 Nijm II —123 (—20) Table 2. Binding energies for the S++ — and sbb — S-b systems obtained with Argonne V18 [7], AV18, Nijmegen II [8] and Paris [9] potentials. The value in brackets corresponds to a second bound state (from [6]). The origin of the large binding energy given by the AV18 interaction model is its large squared spin-orbit interaction, which acts in the D—state. In the large Nc limit the angular momentum dependent interaction components are proportional to one power of 1/Nc for each power of the angular momentum L. As the baryon mass scales as Nc that would suggest that an additional factor of mN /mH, where mN is the nucleon and mH the heavy hyperon mass respectively, should be associated with the each power of L in the scaling relations [11,12]. Inclusion of such a factor would suppress the role of the angular momentum dependent interaction operators, and would reduce the calculated binding energies obtained with the AV18 interaction. The interaction between the ground state multiplets of double-beauty should be similar to that of the double charm hyperons, as the main interaction is that between their light flavor quarks. Their binding energy will however be larger than that of double-charm hyperons in view of their larger mass. This is shown in Table 2 where the binding energy of two double-beauty hyperons is calculated with the assumption that their mass is: Mbbu = Mbbd ~ 2mb « 2 x 4500 MeV. Metastability is in this case is determined by a similar inequality, as in the case of the charmed hyperon case: Ab = Mbbb + Mbu — 2Mbbl < 0, where its estimated value ranges [348 — 372] MeV [10]. Therefore he result obtained with the AV18 potential is a bound state, while that obtained with the Nijm II potential is a metastable state. Deuteron-like bound states of nucleons and double-heavy hyperons: N — Ecc and N — Ebb are also very likely. The AV18 potential gives bound states at —388 MeV and —494 MeV for the N — —cc and N — —bb systems respectively. The Nijm II potential gives bound states at —35 MeV and —76 MeV for the N — Ecc and N — Ebb systems respectively. Acknowledgment This talk is based on collaboration with Bruno Julia-Diaz. Research supported in part by the Academy of Finland under grant 54038. References 1. M. Mattson et al., Phys. Rev. Lett. 89 (2002) 112001; M. Moinester et al., hep-ex/0212029; A. Ocherashvili et al., hep-ex/04506033. 2. S. Fleck and J. M. Richard, Prog. Theor. Phys. 82 (1989) 760. 3. M. Rho, D. O. Riska and N. N. Scoccola, Z. Phys. A 341 (1992) 343. 4. J. M. Flynn, JHEP 0307, 066 (2003) 5. F.Coester, K. Dannbom and D. O. Riska, Nucl. Phys. A 634 (1998) 335 6. B. Julia-Diaz and D. O. Riska, nucl-th/0405061 7. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51 (1995) 38. 8. V. G. J. Stoks et al., Phys. Rev. C 49 (1994) 2950. 9. M. Lacombe et al., Phys. Rev. C 21 (1980) 861. 10. J. M. Richard, Phys. Rept. 212 (1992) 1; B. Silvestre-Brac, Few Body Syst. 20 (1996) 1. 11. D. B. Kaplan and A. V. Manohar, Phys. Rev. C56 (1997) 76. 12. D. O. Riska, Nucl. Phys. A710 (2002) 99. Bled Workshops in Physics Vol. 5, No. 1 Calculation of electroproduction amplitudes in the K-matrix formalism* B. GolliQ'b, P. Albertoc,e, L. Amoreirad-e, M. Fiolhaisc-e, and S. Sircaf-b Q Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia bJ. Stefan Institute, 1000 Ljubljana, Slovenia cDepartment of Physics, University of Coimbra, 3004-516 Coimbra, Portugal d Department of Physics, University of Beira Interior, 6201-001 Covilha, Portugal e Centre for Computational Physics, University of Coimbra, 3004-516 Coimbra, Portugal fFaculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Abstract. The K-matrix approach is applied to the calculation of the multipole amplitudes M 1 +, E i +, and S i + in the A channel within the Cloudy Bag Model. The separation of the amplitudes into the resonant part and the background is presented and discussed. 1 Introduction In our previous work [1] (see also [2]) we presented a method to calculate pion electroproduction amplitudes in the framework of chiral quark models. We derived the expressions for the transition K-matrix and the T-matrix and showed how to separate the resonant part from the background. In the present work we apply this method to the calculation of amplitudes M i E i and S i + in the A(1232) channel. We use the Cloudy Bag Model as a simple example of a chiral quark model. In spite of the known limitations of the model we show that it is possible to reproduce these amplitudes sufficiently well in a broad energy range. We explain how to isolate the resonant parts of the amplitudes and show that these parts are in good agreement with the results extracted from the experiment. 2 Electro-production amplitudes and cross-sections In electro-production, the incoming virtual photon with four-momentum (, ky), — kY = —Q2, and polarization interacts with the nucleon with the third components of spin ms and isospin mt; the final state consists of the scattered pion with four-momentum (^0, k0) and the third component of isospin t and the nucleon with good mS and mt. In the c.m. frame the nucleon momentum is opposite to that of the photon (pion). If the z-axis is oriented in the direction of the incoming photon, the K-matrix for this process can be written as Kyn = -n(¥P(ms,mt;ko, t)|HY|N(mS,m{);ky, m) . (1) * Talk delivered by B. Golli Here ¥P is a principal-value state (see e.g. [3]) while |N(m^ ,mt );ky, |i) stands for the asymptotic (free) states representing the nucleon and the photon. The principal value state can be written in the form1: |VP) = (c4(ko)|N(ms,mt)) + dk x(k°'k) aI(1c)|N(ms,mt)) + cr|R)X , V ' J ^k - x J (2) where a|(k) is the pion creation operator, |N) is the nucleon state, and |R) is a (possible) resonant state with excited internal degrees of freedom (e.g. quarks and/or mesons). The amplitude describing the scattered pion, x(k0, k), is related to the phase shift. The state (2) is normalized as (^(E)^ (E')) = (1 + K2W 5(E - E')5ap , (3) where E is the total energy of the system K is the K-matrix for pion scattering, and a, |3 label different channels. The normalization (3) is not practical in numerical calculations because the factor in front of the 5 function diverges as E approaches the resonant energy. It is more convenient to work with the state normalized simply to 5(E — E') at the resonance: |¥r) = K^|¥P). (4) We now expand ¥P (or equivalently in (1) in states with good total angular momentum J and isospin T: Kyyt = K«« v^k; Y_ J, T, M J Mt; k0, l| [Hy , a], (ky )] |N (m^, )) lm xu^)cf;slmc™;lt+..., (5) where is the angle between the scattered pion and the incident photon, ajjky) is the creation operator for the photon, the factor y/cuyky ensures the proper normalization of the photon asymptotic state, and C's are the Clebsh-Gordan coefficients. Since we are usually interested in one particular channel with given J and T we have denoted by . . . other channels not taken into account. The T-matrix is obtained as Tyn = Ky„(1 + iT„„), (6) yielding a similar expression as (5) in which is replaced by Tnn. The appearance of (Tnn) in front of the (real) transition amplitude means that the phase shift of the transition K or T-matrix is that of the meson scattering - an explicit manifestation of the Watson theorem. In fact, in the above derivations, we have tacitly assumed that "switching on" the electro-magnetic interaction HY does not change the strong scattering amplitudes, i.e. the principal-value state (2) remains unchanged. 1 Here the normalization of the principal value state (see (3)) and consequently the definition of the K-matrix is changed slightly with respect to the ones used in [1]. To obtain the electro-production amplitudes in the A-channel, we keep only the p-wave pions and the J = T component of the final state in (5); we furthermore neglect nucleon recoil and the effect of the two-pion decay channel. The pertinent electro-production amplitudes are related to the matrix elements of the T-matrix, by M (3/2) 1 + 3 1 16k0ky n 1 (3K3/2 + V3K1/2) (7) and E (3/2) E1 + 3 11 16k0ky n (K3/2 — V/3K1/2). (8) Here we have introduced the analogues of the familiar helicity amplitudes: Ka = \/Wy\<^ (Vr(MJ = A)| e0 dr e^ ■ j(r)elkY'r|N(mS = A - , (9) where j (r) is the vector part of the electro-magnetic current. The differential cross section then reads ^ = ^{l|M1 + |2(5-3cos2£) + ?|E1 + |20+cos2i>) + 3ReMl + Ei + (1 - 3 cos2 -&)J . The longitudinal amplitude is 1+ with 1 e0 32n2 k0 dreo • j(r)e T'r|N(mg = 4)), (10) doL k0 dâ ~ k^ |Li +|2{4 + 12 cos2 . (11) 3 2 T 2 3 Calculation of the K-matrix in chiral quark models We consider quark models in which the p-wave pions couple to the three-quark core. Assuming a pseudo-scalar interaction, the pion part of the Hamiltonian is Hn dk^ amt(k)amt(k) + [ymt(k)amt(k) + Vmt(k)t a^t(k)] } , (12) where a^tM is the creation operator for a p-wave pion with the third components of spin m and isospin t, and Vmt (k) = —V(k) Y.3=1 °mTTtr represents the general form of the pion source in which the function V(k) depends on the particular model. In the Cloudy Bag Model, V(k) reads vik) = k2 <11 iiM (13) VU7t2cvk ^k — ^o (14) Here ® a denotes the resonant state representing the bare delta (i.e. three quarks in s-state coupled to J = T = ■§) and a cloud of up to two pions around the bare nucleon and delta. The pion profiles in and ®a can be most easily determined from the following relations that hold for Hamiltonians of the type (12): Vt (k) amt(k)|A)=--ftrV'A> <15) ^k + H — tA and vmt (k) vm ,t, (k') fn >78 MeV for 0.9 fm < R < 1.1 fm, respectively. As seen from Figs. 2 the experimental values for the electroproduction amplitudes are underestimated. The reason lies in a too weak yNA vertex, which is a known feature of the Cloudy Bag Model. Taking a smaller R and reducing further the value of fn [4] enhances the contribution of the pion cloud, and thus increases the strength of the yNA vertex. Yet this mechanism does not help to improve the agreement: increasing the strength of the quark-pion interaction leads to a larger width of the resonance, and since VT appears (implicitly) in the denominator of the amplitudes (9) and (10), the net effect is such that the magnitude of the Im Mi + in the vicinity of the resonance decreases. 3 3 T 7 4 Extracting the resonance In some models, the delta resonance is described as a particle with a finite lifetime and an energy corresponding to the pole of the T-matrix in the complex energy plane. The properties of such a particle can not be directly related to the measured amplitudes since the amplitude include also non-resonant processes. In this section we show how to relate the results obtained in the K-matrix approach to those of the above mentioned models. The resonant part of the amplitudes is usually assumed to have a Breit-Wigner shape with a constant width (see Eq. (18)) below). In order to identify the part in the total amplitude that possesses this type of behavior we write the pertinent K-matrix in the form proposed in [5]: C KytTt = 7-r + D - (17) Ea — E with two constant coefficients C and D. Using these two parameters and the experimental value for EA we obtain an excellent fit to the calculated phase shift (see Fig. 1). The corresponding T-matrix can be cast in the form, suggested by Wilbois et al. in the speed-plot analysis (Eqs. (71)-(76) of [6]): TytTt = T= e216* T]/2.rln + sin5bel6* . (18) 1 - iK„„ MA - E - irA/2 The parameters of the T-matrix can be easily deduced from (17) and are given in Table 1. Since we started from a real K-matrix, the resulting T-matrix automatically obeys unitarity, which is an important merit of our approach. Fig.1. The phase shift in the P33 channel as a function of the invariant mass. The data points are the single-energy values of the SM02K (2GeV) solution of the SAID nN partial-wave analysis [7]. The thick line represents the calculated phase shift, while the thin line is the two-parameter fit to the calculated values. The agreement is worse only above 1300 MeV where the two-pion channel becomes relevant and our approach is not valid anymore. W [ MeV] In a similar way we can split the K-matrix for the electroproduction in the resonant and the background part: A Kt« = ---+B. (19) t A — t The parameters A and B for each multipole can be determined by fitting the calculated amplitudes using the form implied by (7) and (8): M = -jL= , Ky" , (20) Y'koky 1 — iKnn where M is ether m1+2) or e1+2) . Alternatively, one can use a simplified form: ^ = (21) I — i Knn which is more frequently used in the experimental analysis, e.g. in [8] and in the SP-analysis of [6]; the form (20) being used in the MSP analysis of [6]. The resulting parameters are listed in Table 1. Table 1. Resonance pole parameters extracted from the computed phase shifts and electro-production amplitudes using the form (21). Parameter C is the resonance width divided by 2, D is the tangent of the background phase shift, and Ma and rT are parameters of the T-matrix (see (18)). Experimental values are the recent PDG values [9] and from [6]. R fn C D Ma Tat A(M1) B(M1) A(E2) B(E2) [fm] [MeV] [MeV] [MeV] [MeV] [10-3/m«] [10-3/m„] 1.1 78 57 -0.39 1213 49 0.0123 -2.57 -0.000235 -1.19 1.0 81 56 -0.40 1213 48 0.0117 -3.53 -0.000236 -1.09 0.9 83 56 -0.41 1212 48 0.0115 -4.00 -0.000221 -1.00 Experiment 60 -0.435 1210 50 1 100 1200 1300 W [ MeV] -2 - ,iT||f ' \ 0 / t % '%/* / * 1 Ti,, 1, L + i ■> r -w '| T Re - 't - 1 100 1200 1300 W [ MeV] Fig.2. The and the electro-production amplitude in the CBM by using R = 1.0 fm and fn = 81 MeV. The data points in the figures are the single-energy values of the SM02K (2 GeV) solution of the SAID nN partial-wave analysis [7]. The thick lines represent the calculated amplitudes for R = 1.0 fm and fn = 81 MeV, while the thin lines are the fits to the calculated values using the parameters from Table 1. From our results it is possible to extract the resonance parameters at the pole of the T-matrix, based on the separation of the amplitude into the resonant and 68 B. Golli, P. Alberto, L. Amoreira, M. Fiolhais and S. Sirca background parts, using the parameterization [6,8] T = Tr + Tb , Tr rr// ei^ M/ E - ir//2 Using (20), the parameters r, and TB can be expressed in terms of A, B, C, and D. The moduli and phases for the transverse multipoles are shown in Table 2 together with the EMR ratio. While the magnitudes are underestimated, the ratio as well as the phases are much better reproduced. Table 2. Resonance pole parameters extracted from the computed E13/2) and M1 + z 1 multipoles using the form (21) and parameters in Table 1, compared to various determinations from data. The moduli r are in units of 10-3 /mn. Ra is the EMR ratio at the pole of the T -matrix. ,(3/2) R [fm] / f n [MeV] te tm ^ m Ra 1.1 / 78 0.75 -154° 16 —25° -0.031 - 0.037 i 1.0 / 81 0.72 -158° 15 —28° -0.030 - 0.037 i 0.9 / 83 0.67 -159° 14 -31° -0.029 - 0.037 i Ref. [8] 1.23 -154.7° 21.2 -27.5° -0.035 - 0.046 i Ref. [6] (SP) 1.23 -156° 19.9 -26.0° -0.040 - 0.047 i Ref. [10], Fit 1 1.22 -149.7° 22.2 -27.4° -0.029 - 0.046 i Ref. [11], Fit A 1.38 -158° 20.9 -31 ° -0.040 - 0.053 i Table 3. Same as Table 2 except that the parameterization (20) is used. R [fm] / f n [MeV] re rm ^m Ra 1.1 / 78 0.74 -157° 16 -34° -0.026 - 0.038 i 1.0 / 81 0.68 -160° 15 -37° -0.025 - 0.037 i 0.9 / 83 0.62 -162° 14 -40° -0.023 - 0.037 i Ref. [6] (MSP) 1.12 -162° 20.7 -36.5° -0.032 - 0.044 i 5 Discussion We have presented a method to calculate directly the K-matrices of resonant electro-production processes in the framework of chiral quark models. The identification of the resonant part and the background is unambiguous in the K-matrix formalism. In the T-matrix formalism, this separation is based on the assumption that the position and the width of the resonance do not depend on the invariant energy and is intimately connected to our picture of a resonance as a short-lived particle. While such an assumption cannot be justified in a microscopic model, it is surprising how well it reproduces the experimental results in a broad range of energies. (The agreement at low and high energies in Fig. 2 can be improved by assuming that the background part is energy-dependent.) Fig. 3. Two processes dominating the P33 channel Neither the resonant part nor the background are related to a specific process, such as those depicted in Fig 3. Naively, one would expect that graph (b) corresponds to the resonant part and graph (a) to the background. Yet they both contribute to the resonant part as well as to the background; note that the process (a) alone can lead to the resonance in this channel for sufficiently strong nN coupling and has the opposite sign with respect to the background contribution in the whole energy range. Let us conclude by noting that a good microscopic model should be able to reproduce the total amplitude and not just the resonant part, since, as seen from Tables 2 and 3, the extracted values from the experiment are too unreliable to serve as benchmarks. References 1. B. Golli, in: B. Golli, M. Rosina, S. Sirca (eds.), Proceedings of the Mini-Workshop "Effective Quark-Quark Interaction, July 7-14, 2003, Bled, Slovenia, p. 83. 2. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, S. Sirca, submitted to Phys. Lett. B, hep-ph/0409246. 3. R. G. Newton, Scattering Theory of Waves and Particles, Dover Publications, New York 1982. 4. K. Bermuth, D. Drechsel, L. Tiator, J. B. Seaborn, Phys. Rev. D 37 (1988) 89. 5. R. M. Davidson, N.C. Mukhopadhyay, Phys. Rev. D 42 (1990) 20. 6. Th. Wilbois, P. Wilhelm, H. Arenhovel, Phys. Rev. C 57 (1998) 295. 7. R. A. Arndt, W. J. Briscoe, R. L. Workman, 1.1. Strakovsky, SAID Partial-Wave Analysis, http://gwdac.phys.gwu.edu/. 8. O. Hanstein, D. Drechsel, L. Tiator, Phys. Lett. B 385 (1996) 45. 9. S. Eidelman et al., Phys. Lett. B 592 (2004) 1. 10. R. M. Davidson et al., Phys. Rev. C 59 (1999) 1059; the average of VPI and RPI analysis results is listed. 11. R. Workman, R. A. Arndt, Phys. Rev. C 59 (1999) 1810. Bled Workshops in Physics Vol. 5, No. 1 Molecular binding of Tcc = DD* tetraquark * Damijan Janca and Mitja Rosinaa,b bJ. Stefan Institute, 1000 Ljubljana, Slovenia Q Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Abstract. We present the results of detailed calculations with Bhaduri and AL1 potential for the Tcc = DD * tetraquark. We show that it has a molecular structure, which can transform, under the influence of an additional three-body force, into a Ab-like system where the role of the b quark is played by the heavy cc diquark. Nonrelativistic potential models have proven to be quite a successful tool for understanding the meson and baryon sector. It is challenging to extend them from one-hadron to two hadron systems, such as the double heavy tetraquarks. Probably the most intriguing tetraquark in this class is the Tcc = DD* tetraquark. The results obtained with different potential models are very contradictive, from unbound [1,2] to deeply bound states [3,4]. If one demands, however, that the model used in the calculations must reproduce accurately the meson as well as baryon sector, then we believe that the dependence of the results on the model should not be so strong. Actually, the results should only be sensitive to the details of the interaction, which are not of the great importance for the meson or baryon sector, such as for example the colour dependent three-body force. We present the results obtained with two one-gluon-exchange potentials, the Bhaduri [5] and Grenoble AL1 [6] potential. For a long time it was supposed that Tcc is unbound with these two potentials, according to seemingly accurate calculations [2,7]. We expanded the tetraquark wavefunction in 140 Gaussians of optimized widths for three sets of Jacobi coordinates to obtain 0.1 MeV accuracy (Fig.1) and show [8,9], however, that with both, the Bhaduri and the Grenoble AL1 potentials, Tcc is bound below the DD* threshold by 0.6 and 2.7 MeV, respectively. It is essential to use a large enough model space to accommodate the molecular structure, in contradistinction to Tbb which has an atomic structure similar to Ab. Both types of configurations are schematically illustrated in Fig.2. If the basis is too small the Tcc tetraquark without additional interactions remains unbound. This had happened in [10], where the same basis functions were used as here, but the final basis was spanned with only 40 functions, since so extremely weak binding was not expected. From Fig.1 we see that at least 80 basis function are needed to obtain the energy of the DD* system lower than the threshold. * Talk delivered by D. Janc. 3910 0 20 40 60 80 100 120 140 N Fig.1. Energy of the Tcc tetraquark with Bhaduri potential as a function of the number of the basis states for three different runs. The D + D* threshold is also shown. Since the initial parameters are chosen randomly, the convergence is similar as with the stochastic variational approach. 3909 3908 3907 3906 3905 In Fig.3a we plot the probability densities Pqq between heavy quarks in Tbb and Tcc as a function of the interquark distance rQQ: Pqq(r) = (^i§(r - Tqq)|^). a) b) Fig. 2. Schematic illustration of the two light antiquarks (empty circles) and two heavy quarks (dashed circles) in a): atomic configuration as we can find it in the Tbb tetraquark and in b): molecular configuration characteristic for the Tcc tetraquark. There are also other mechanisms to help binding: 3-body forces (which are more effective for 4 particles than for 3 particles - baryons) and pion exchange (pions are almost real when exchanged between D and D* mesons). The form of the three-body interaction which we introduced into the tetraquark is Vq3bqq (n, rj, rk) = AfA^Uo expH^ + ifk + r^/r2], V^q(ri, rj.ru) = ldttbcAf Ab*A^U0 expH^ + r?k + r2, )/r£]. Here r^ is the distance between i-th and j-th (anti)quark, and similarly for rjk and rki. Aa are the Gell-Mann colour matrices and dabc are the SU(3) structure constants ({Aa, Ab} = 2dabcAc). It should be noted that in the baryon sector such a colour structure is irrelevant since there is only one colour singlet state and thus the colour factor 72 D. Tanc. M. Rosina Fig.3. (a): Tbb shows atomic structure while Tcc is molecular, r = rbb or rcc; (b): The effect of three-body interaction on the structure of Tcc for 3 different strengths. is just a constant which can be included into the strength of the potential. In tetraquarks the situation is different since there are two colour singlet states: 312334 and 612 634 (or li3124 and 813824 after recoupling). The three-body force operates differently on these two states [11,12] and one can anticipate that in the case of the weak binding it can produce large changes in the structure of the tetraquark. This cannot be otherwise produced simply by reparameterization of the two-body potential, so the weakly bound tetraquarks are a very important laboratory for studying the effect of such an interaction. It is well known that the constituent quark models with the colour A ■ A structure give rise to the long-range van der Waals forces [13-15], which can have dramatic effect especially for weakly bound systems with the molecular structure, such as the Tcc tetraquark. This interaction appears due to the colour polarization of two mesons in the colour singlet state and is an artefact of the potential approach. It is not present in the full QCD where quark-anriquark pair creation from the confining filed energy would produce an exponential cut-off of this residual interaction. The radial dependence has in the case of the linear confining interaction the structure V (r)v.d.Waais = O(rd-4 ) = O(r-3) We now check the effect of this spurious interaction in the Tcc tetraquark. In Fig. 4 we present useful quantity, which we call effective potential density vij (r) = MVj (rtj )5(r - ry )|^) = Vj (r)Pij (r). (1) In Fig. 4b one can see that this effect is indeed present at large separations (r > 2 fm) but is extremely small. Integrating this attractive tail of the potential, we obtain a contribute less than 100 keV to the binding of the system. On the other hand, more interesting feature of the effective potential shown in Fig. 4 is the repulsive force between quarks at the medium distance between quarks (1.5 fm> r > 2 fm). The maximal value of potential barrier is Vy (r ~ 1.5 fm) = Vij/Pij = 1 MeV. This then allows that also resonant states can appear in the model which are not possible in a simple potential well. Fig. 4. Left: Potential densities vij between (anti)quarks as calculated from Eq. 1 for Bhaduri potential. Right: Enlarged section of the left-hand side figure, where van der Waals attraction and medium-range repulsion can be seen. References 1. Manohar A. V., Wise M. B.: Nucl.Phys. B399,17 (1993). 2. Silvestre-Brac B., Semay C.: Z. Phys. C57, 273 (1993). 3. Vijande J., Fernandez F., Valcarce A., Silvestre-Brac B.: Eur. Phys. J. A19, 383 (2004). 4. Pepin S., Stancu Fl., Genovese M., Richard J.M.: Phys. Lett. B393,119 (1997). 5. Bhaduri R. K., Cohler L. E., Nogami Y.: Nuovo Cim. A65, 376 (1981). 6. Silvestre-Brac B.: Few-Body Systems 20,1 (1996). 7. Semay C. Silvestre-Brac B.: Z. Phys. C61, 271 (1994). 8. Janc D., Rosina M.: hep-ph/0405208. 9. Del Fabbro A., Janc D., Rosina M., Treleani D.: hep-ph/0408258. 10. Janc D., Rosina M.: Bled Workshops in Physics 4, No.1, 89 (2003). 11. Dmitrasinovic V.: Phys. Lett. B499, 135 (2001). 12. Dmitrasinovic V.: Phys. Rev. D67, 114007 (2003). 13. Weinstein J., Isgur N.: Phys. Rev. D27, 588 (1983). 14. Greenberg O.W., Lipkin H.J.: Nuc. Phys. A370, 349 (1981). 15. Feinberg G., Sucher J.: Phys. Rev. D20,1717 (1979). Bled Workshops in Physics Vol. 5, No. 1 New ideas about production and detection of cc-tetraquarks * Mitja Rosinaa,b and Damijan Janca bJ. Stefan Institute, 1000 Ljubljana, Slovenia Q Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Abstract. We estimate the rate of double charm production in B-factories Belle and BaBar, in hadronic machines with fixed targets RHIC and SELEX, and in high energy colliders Tevatron and LHC. For detection we propose the branching ratio between pionic and gamma decay. 1 Introduction We have shown that the molecule-like configuration of the DD* dimeson (also called tetraquark) enables weak binding even in the case of the Bhaduri or Grenoble AL1 interaction (-0.7 or -2.6 MeV, respectively) [1,2]. The surprise that the cc-tetraquark is likely to be bound against the D + D * decay strongly motivates experimental exploration. To encourage future experimental analyses, we estimate the production rate on several present and future machines, and propose an experimental signature for detection. 2 Production Regarding the production of Tcc, we consider a three-step process: (i) production of two cc pairs, (ii) formation of a diquark c + c —> cc, (iii) dressing of the diquark cc —> ccq, q = u, d, s (90 %), or cc —> ccud (10 %). Here are some provocative guesses: [3] • SELEX [4] has seen 50 candidates for ccq the corresponding hypothetical 5 Tcc are to few to be recognized at present. • Belle reported prompt J/i[> production in e+e~ annihilation at y7? = 10.6 GeV and found that the most of the observed J/^ production is due to the double cc production a(e+e- —> J/^cc)/o-(e+e- —> J/^X) = 0.59 which correspond to 2000 events from their 46.2 fb-1 data sample promising for the Tcc production! Similar rate is also expected for BaBar. * Talk delivered by M. Rosina. • High energy colliders (RHIC (p-p), RHIC (p-Au); Tevatron, LHCb, LHC--ALICE) might produce sufficient double charm by double two-gluon fusion [5-8,3] (g + g) + (g + g) —> (c + c) + (c + c). Our estimate for the Tcc cross section are 4, 750; 21, 27,58 nb, respectively. In most machines, the rate seems promising to start the hunt! 3 Detection The main problem with detection of the weakly bound Tcc tetraquark is how to distinguish the pion or photon emitted by the decay of the free D * meson from the one emitted by the D* meson bound inside the tetraquark. We can exploit the fact that the phase space for D * —> D + n decay is very small. Therefore we propose as a signature the branching ratio between radiative and pionic decay. In addition, the analysis using the Dalitz plot can help to distinguish whether the pion was emitted from a bound state, resonance state of D + D * or from free D * meson. 4.04 r 4.03- > 4.02- DDn decay, E is binding energy of Tc Right: The two graphs contributing in the case of resonance E > 0 D n D D D 3 is pion; 4 References 1. These Proceedings 2. D. Janc and M. Rosina, hep-ph/0405208, submitted to Few-Body Systems. 3. D. Janc and M. Rosina, hep-ph/0408258, submitted to Phys. Rev. D. 4. M. Mattson et al. (SELEX Collaboration), Phys. Rev. Lett. 89, 112001 (2002); J. S. Russ (on behalf of the SELEX Collaboration), hep-ex/0209075 5. A. Del Fabbro and D. Treleani, Phys. Rev. D61, 077502 (2000); Phys. Rev. D63, 057901 (2001); Nucl. Phys. B 92,130 (2001). 6. M. Rosina, D. Janc, D. Treleani, and A Del Fabbro, Bled Workshops in Physics 3, No.3, 63 (2002). 7. D. Janc, M. Rosina, D. Treleani, and A. Del Fabbro, Few-Body Systems Suppl. 14 (2003) 25. 8. A. Del Fabbro and D. Treleani, Bled Workshops in Physics 4, No.1, 75 (2003). Bled Workshops in Physics Vol. 5, No. 1 Recent results on A resonance production at MIT-Bates, MAMI, and JLab (Hall A) S. Sircaa,b a Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia bJozef Stefan Institute, 1000 Ljubljana, Slovenia Abstract. Electro-production of mesons on nucleons is the optimal tool to investigate the dynamics of nucleon resonance excitation. In the past years, tremendous advances have been made based on new instrumental capabilities of modern electron beam facilities, in particular by measuring polarization observables. Some of the recent results on A resonance production from three major coincidence electron-scattering collaborations are presented. 1 The facilities Modern electron-scattering facilities possess distinct instrumental features which allow for a mutually complementary kinematic coverage, exploitation of various polarization degrees of freedom (e.g. through measurement of double-polarization observables), and different controls of systematic uncertainties. The MIT-Bates facility has two collaborations: the Out-of-Plane Spectrometer System (OOPS) and the Bates Large-Acceptance Spectrometer Toroid (BLAST). Both utilize ~ 1 GeV polarized electron beams of the Bates linac, in extraction (quasi-CW) or storage mode, respectively. OOPS has recently stopped taking data and is now in the process of data analysis. It operated four relatively light-weight spectrometer modules that can be positioned almost independently about the momentum transfer direction, and out of the electron scattering plane, to detect protons and charged pions [1]; this ensures an excellent control of systematics. BLAST is a large-acceptance toroidal magnetic spectrometer [2] that has only recently started taking production data, with a capability of simultaneous detection of charged and neutral particles in large momentum and angular ranges, with a moderate energy resolution. Its key feature are the gaseous, isotopically pure, vector-polarized hydrogen, and vector- and tensor-polarized deuterium internal targets. In a high-luminosity environment of the MIT-Bates storage ring, excellent figures of merit are achievable, which enable us to access double-polarization observables in a number of physical channels. The A1 Collaboration at the MAMI-B accelerator makes use of the high-polarization, ~ 0.9GeV CW beam in conjunction with either target (high-polarization 3 He) or recoil polarimetry (focal-plane polarimeter), and a setup of three high-resolution spectrometers [3] (one of them can be positioned out of plane). In addition, individual dedicated spectrometers or non-magnetic detector systems are installed periodically for measurements of specific reaction channels. The accelerator is presently being upgraded to the energy of 1.5 GeV, and one of the spectrometers is being added to the setup to accommodate the higher particle momenta. The Hall A Collaboration at Jefferson Lab operates two high-resolution magnetic spectrometers and auxiliary detector systems, making use of the high-polarization CW beam of energies up to 6 GeV. Both target polarization (3He with similar operational parameters as at A1) and recoil polarimetry (focal-plane po-larimeter with optimizable secondary-scattering configuration) are possible. The large kinematic freedom given by the high beam energies allows us to explore the nucleon resonance production at relatively high Q2, with invariant energies W extending beyond - 2 GeV. 2 Pion-cloud effects at low Q2 One of the key goals of the experiments devoted to the N —> A transition is to determine the electric (E2) and Coulomb (C2) quadrupole transition amplitudes. These are much smaller than the leading magnetic dipole amplitude (M1), and indicate that the nucleon and/or the A deviate from spherical symmetry. In models involving explicit pion degrees of freedom, large contributions to M1 and dominant contributions to E2 and C2 can be attributed, schematically, to the pion cloud surrounding the bare quark core (or pion loop effects). The motivation behind the recent N A program at MIT-Bates and MAMI is therefore to map out the M1, E2, and C2 multipoles in the region of low Q2 ~ 0.1 (GeV/c)2 where pion-cloud effects are expected to play the most important role. The electric quadrupole amplitude E2 is accessible through a particular combination of the partial cross-sections ac„(en) = oo (en) + ail (en) - oo (180°) - 2 (cos en + 1) Re [E0+ Mi +] - 12 sin2 en Re [E1+ Mi + ] , where en is the center-of-mass emission angle of the pion and o0 = + £oL. It is clear that o0n exhibits a large sensitivity to EMR - Re [E|+ M1 + ]. However, backgrounds like the electric dipole amplitude E0+ in the Re [E0+M1+ ] interference, as well as higher partial waves (I > 2), need to be obtained from a model in order to extract the EMR. Similarly, the quadrupole amplitude C2 is accessed through LT-terms in the cross-section which contain interferences of the scalar quadrupole S1 + with the dominant magnetic dipole M1 +: OLi (en) - sin en Re [S0+ M1 + ] - 6 cos en sin en Re [S|+ M1 + ] , olt' (en) - - sin en Im [(-6 cos en S1+ + So+ )* M1+ ] . The oLT is primarily sensitive to CMR - Re [S|+M1+] while oLT', accessible only with a polarized beam and out-of-plane detection, probes Im [S|+M1+]. (This is important as the relative phases between the multipoles need to be fixed.) The analysis of all existing OOPS data at Q2 = 0.127 (GeV/c)2, including the latest runs with the CW beam at MIT-Bates [5], yield EMR = (-2.3 ± 0.3stat+sys ± 0.6 model) % i CMR = (-6.1 ± 0.2stat+sys ± 0.5model) % . At this moment, these are the most accurately known EMR and CMR values at any finite value of Q2. (Note that the E2 multipole and EMR are more difficult to isolate in electro-production than C2 and CMR because the transverse responses are dominated by |Mi +12 which is absent in the longitudinal sector.) The extracted CMR is in agreement with the older OOPS extractions, with the Mainz determination from recoil polarimetry at Q2 = 0.121 (GeV/c)2 which resulted in CMR = (-6.4 ± 0.7stat ± 0.8sys) % [6], as well as with the CLAS data in a broader Q2-range [7]. (New preliminary results for EMR and CMR from CLAS exist at Q2 up to 6 (GeV/c)2 and have been reported at various meetings in 2004.) In addition to the extractions of EMR and CMR at low Q2, the present data sets will be used to try to answer several open questions arising from previous experiments at MIT-Bates and MAMI (see contribution of S. Sirca to the 2003 Proceedings [8]). When final results in ctlt, ctlt', and other partial cross-sections from OOPS and MAMI become available, they will help constrain the models of pion electro-production [9-11]. In particular the observables involving polarized beams in conjunction with either polarized targets or recoil polarimetry, represent severe tests of the models. Preliminary results on ctlt ' from the MAMI runs in 2003 are shown in Fig. 1. Preliminary W=1221 MeV, Q2=0.060 GeV2 Fig. 1. Preliminary results on ffLT' ~ Im [S1+ Mi + ] from MAMI, compared to three state-of-the-art model calculations [9-11]. 3 Multipole decompositions at high Q2 To minimize the model dependence of the extracted multipole amplitudes, a measurement with a sufficient number of independent observables is needed. The N a transition cross-section in the case of a polarized beam, unpolarized target, and recoil polarimetry, can be decomposed into 18 independent structure functions, each one of which contains different forms of multipole bilin-ears. Through a partial-wave analysis of the measured angular distributions of the structure functions, all relevant multipoles can be extracted from the data in a model-independent way. By measuring the angular distributions of 16 independent structure functions in broad angular ranges, the Hall A experiment E91-011 has succeeded in delivering Re and Im parts of all i = 0,1 multipoles in the vicinity of q2 = 1.0 (GeV/c)2 and w = 1232MeV. The residual model-dependence is due to the higher partial waves (i > 2) which were constrained by MAID. WvsQ2 Fig. 2. Kinematical coverage in the E91-011 experiment, with indicated binning for the polarization analysis. Left: angular acceptance in recoil nucleon center-of-mass angles; Right: acceptance in W and Q2. Recoil polarimetry in the pn0 channel is indeed the most powerful and hence the preferred method to cleanly disentangle individual multipoles; however, this goal could be achieved because of the strong kinematic focusing of the proton emission cone into the spectrometer acceptance at relatively high q2. In this way, a substantial angular coverage was achieved (see Fig. 2). The measured structure functions at w =(1.23 ± 0.02) GeV and q2 = (1.0 ± 0.2) (GeV/c)2 are shown in Fig. 3. The final analysis which will result in the individual multipoles, as well as the EMR and the CMR is almost complete, and will be reported soon. Fig. 3. Preliminary E91-011 results for the polarized structure functions in p(e, e 'p)rt° at W = (1.23 ± 0.02) GeV and Q2 = (1.0 ± 0.2) (GeV/c)2, compared to the pion electro-production models, and different multipole fits. 4 Work in progress and outlook The analysis of the data taken with the OOPS spectrometer system at Q2 = 0.127 (GeV/c)2 is underway both in the pn0 and the nn+ channels, at the resonance (W = 1232 MeV) and below it (W = 1175 MeV). Selected unpolarized responses have measured which allow for a precise extraction of the EMR and CMR ratios with a relatively small model dependence. By measuring two channels, a first step towards the isospin decomposition of the amplitudes will have been made. Preliminary responses in the pn0 channel from A1 at MAMI are already available, while the full analysis is expected to be complete soon. We expect it to yield five unpolarized responses and the EMR and CMR ratios at Q2 = 0.06 and 0.2 (GeV/c)2, where the effects of the pion cloud appear to be most prominent. The measurement of o^r alone, with respect to the older A1 [12] and the latest CLAS (JLab) [13] data set, will represent an important constraint on the state-of-the-art models, in particular by constraining the I = 0 background amplitudes. (In oLr, the discrepancies between the theories in the I = 0 partial waves arise predominantly through the Im [M1+S0+] interference.) The data analysis of the N A experiment in Hall A has been concluded and is being prepared for publication. The focal-plane polarimetry approach used in this experiment can be straightforwardly extended to the energy region of the Roper resonance; an experiment proposal is presently being considered. However, the cross-sections in the second resonance region are far smaller than in the A region, and the sensitivities to the resonant Roper multipoles appear to be largest at small Q 2 where the kinematic focusing is too weak to allow for a full partial-wave decomposition. References 1. Z.-L. Zhou, S. Sirca et al., Nucl. Instr. and Meth. A 487 (2002) 365. 2. R. Alarcon, Prog. Part. Nucl. Phys. 44 (2000) 253. 3. K. I. Blomqvist et al., Nucl. Instr. and Meth. A 403 (1998) 263. 4. J. Alcorn et al., Nucl. Instr. and Meth. A 522 (2004) 294. 5. N. F. Sparveris et al., submitted to Phys. Rev. Lett., arXiv:nucl-ex/0408 003. 6. Th. Pospischil et al. (A1 Collaboration), Phys. Rev. Lett. 86 (2001) 2959. 7. K. Joo et al., Phys. Rev. Lett. 88 (2002) 122001. 8. S. Sirca, in: B. Golli, M. Rosina, S. Sirca (eds.), Proceedings of the Mini-Workshop "Effective Quark-Quarklnteraction, July 7-14, 2003, Bled, Slovenia, p. 107. 9. D. Drechsel, O. Hanstein, S. S. Kamalov, L. Tiator, Nucl. Phys. A 645 (1999) 145. 10. T. Sato, T.-S.H. Lee, Phys. Rev. C 63 (2001) 055201. 11. S. S. Kamalov, S. Yang, Phys. Rev. Lett. 83 (1999) 4494. 12. P. Bartsch et al. (A1 Collaboration), Phys. Rev. Lett. 88 (2002) 142001. 13. K. Joo et al., Phys. Rev. C 68 (2003) 032201(R). Bled Workshops in Physics Vol. 5, No. 1 Search for Pentaquarks at HERA-B Tomi Zivko for the HERA-B Collaboration Abstract. A search for ©+ and S3/2 pentaquarks has been performed in channels p K§ and z. 7tin proton - nucleus interactions at mid-rapidity and y/s = 41.6 GeV/c2 . No evidence for pentaquarks has been found in analyzed channels. Upper limits have been set on pen-taquark production cross sections. Experimental evidence for a new hadron state at 1540 MeV/c2 decaying to nK+ was presented by experiment LEPS [1] in 2003. The particle was named ©+ (1540). Due to a quark picture of neutron and K+, the hadron 0+ must contain at least four quarks and one antiquark. After that, several other collaborations reported evidence for a peak in the invariant mass spectrum of n K+ or p K^ . The p K° peak was regarded as evidence for 0+ —> pKS on the grounds "no narrow !*+ is known around 1.5 GeV/c2 ". Currently, there are 12 experiments that have claimed evidence for decays 0+ —> nK+ or 0+ —> pK^ . The measured mass lies in the range 1521 - 1555 MeV/c2 . There is a peculiarity that p K^ experiments report smaller value of mass than n K+ ones. The measured widths have all been consistent with the experimental resolution which is typically 20 MeV/c2. The presented peaks have statistical significance of about 5 ct. In theoretical models 0+ is a member of an antidecuplet which also contains isospin 3/2 family E3/2 of doubly strange pentaquarks. Evidence for doubly charged and neutral member of the family was observed in En decay channels at mass of 1862 MeV/c2 by NA49 [2]. The statistical significance of E3/2 peak is also about 5 ct. Up to now, this has been the only evidence for E3/2. From the other side, the number of high statistics experiments reporting negative search results for 0+ and E3/2 is growing. Direct comparison of positive and negative search results is not possible because the experiments are not of the same type. However, the negative search results reported much larger yield of common particles like A (1520) and E(1530)0 , thus proving their ability to search for possible pentaquark signals in channels p K^ and E n. This short survey of experimental situation suggests that existence of pentaquarks is not proven beyond reasonable doubt. The search for 0+ and E3/2 was done also at HERA-B. The main features of the analysis are presented here, while details can be found elsewhere [3]. HERA-B is a fixed target experiment at the 920 GeV proton storage ring of DESY. It is a forward magnetic spectrometer with a high resolution vertex-ing and tracking system and good particle identification. The detector has good acceptance in the mid-rapidity region. The informations from the silicon vertex detector, the main tracker system, ring imaging Cherenkov (RICH) counter and the electromagnetic calorimeter (ECAL) were used in this analysis. The present study was performed on a sample of about 200 millions of minimum bias events that were taken at ^/s = 41.6 GeV/c2 using carbon, titanium and tungsten targets. Strange particles are frequent in proton - nucleus interactions at this energy, and HERA-B has reconstructed a large number of Kg —> n+n-, A —> pn- and A —> pn+ decays. A clean sample of E hyperons was obtained in decay modes E- —> An- and EE + —> An+. Background in all fore mentioned channels was efficiently reduced using decay topology, so there was no need for particle identification. Table 1 summarizes the statistics of relevant signals together with the measured mass resolutions. All measured masses are within 1 MeV/c2 compatible with the table values. Signal C target all targets ff [MeV/c2 ] Ks 2.2M 4.9M 4.9 A [A] 440k[210k] 1.1M[520k] 1.6 A (1520) [A(1520)] 1.9k[1.1k] 5.1k[2.3k] 2.3 [2 + ] 4.7k [3.4k] 11.8k [8.2k] 2.6 2(1530)° [2(1530)0 ] 610 [380] 1.4k [940] 2.9 Table 1. Statistics and experimental mass resolution (ff) for relevant particles are given for carbon target and for all targets. HERA-B does not have capabilities for the identification of neutrons. Therefore, the search for 0+ was performed in the decay channel p Kg . Protons were identified requiring the proton likelihood from the RICH to be larger than 0.95. Probability that a particle which is not proton passes this cut is below 1 %. Both particles, proton and Kg had to point to the main vertex. Kg candidates had to lie in ±3ct mass window around the table mass. A clean Kg sample remained after removing particles whose mass was consistent with A or A. The invariant mass spectrum of selected p Kg pairs is shown for p+C data in Fig. 1a. The shape of background was obtained by event mixing technique and is represented by a full line. MC studies show that the mass resolution in the presented mass region is in 2.6 - 6.1 MeV/c2 range. At the 0+ mass, the resolution is 3.9 MeV/c2 . We determined the upper limit on the number of signal events in the invariant mass plot as a function the signal mass. The resulting nuclear cross section as a function of the signal mass is presented in Fig. 1b (full line). Assuming A0 7 dependence of the nuclear cross section on the atomic number, we obtained the upper limit on Br x do/dy|y=0 < 3.7 |b/nucleon in the mid-rapidity region for 0+ mass of 1530 MeV/c2 . The upper limit varies from 3 to 22 |b in the mass region 1521 - 1555 MeV/c2 . The upper limits obtained using data from all targets are similar. We also tried with other search strategies, 50 0 e T3 oq a) , ■ ./...... i i i i i i i i i i i p + C " ........... ■ b) - A * , , , , i , , , i i i i i i i i i i i . iU 11 . i .... i ... . i i i i I i i i i I i i i i I i i i i ■ c) mult < 10 - - ^/fcA^^ ■ }t t p + CT -. +______1___________ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ■ d) mult < 10 + K- all targets: (^ VMh'KW . . .................. 1.475 1.55 pKS mass, GeV/c 1.625 2 > tB 5 m p e ■O 05 500 400 300 200 100 100 20 0 — 1.4 1.6 1. 2 2.2 mass, GeV/ c Fig. 1. Invariant mass distributions and upper limits on nuclear cross section for channels p KS (left) and Sn(right). Arrows denote mass region 1521 - 1555 and mass of 1530 MeV/c2 (left) and mass of 1862 MeV/c2 (right). Data were taken with carbon target. See text for details. 100 0 0 f) 50 0 0 20 like: a) requiring a low track multiplicity in an event (Fig. 1c), b) strangeness tagging, by requiring a particle with an s quark (A, K- ) in an event, c) combination (Fig. 1d) of criteria a) and b), d) relaxation of the proton identification cut. None of the attempts resulted in a significant narrow peak in the mass spectrum. We checked capabilities of the HERA-B detector by reconstruction of A(1520) —> pK-. Masses of ©+ and A (1520) are similar as well as geometrical acceptances for 0+ pK° and A(1520) pK-. Using RICH likelihood cut for both proton and K-, we obtained a clean signal for A (1520). Assuming Br(0+ —> pK^ )=1/4, we determined the UL(95%) on the particle ratio A(f52o) < 0.92% in the mid-rapidity region. This upper limit is more than one order of magnitude lower than predictions of statistical hadronization models. We also found that 1< 0.27%. We searched for members of E3/2 family in decay channels E-n-, E-n+ and c.c. E- candidates had to lie in ±3ct mass window around the table mass. Both E- and n candidates had to point to the main vertex. Weak identification cuts with RICH and ECAL removed tracks with clear electron, kaon or proton identity from the n sample. The invariant mass spectra of En pairs obtained from p+C data are shown in Fig. 1e for all four charge combinations. The background shape is obtained from event mixing and is normalized to the data. The experimental resolution in the analyzed mass region is in 2.9 - 10.6 MeV/c2 range and has value of 6.6 MeV/c2 at the mass of 1862 MeV/c2. The only observed structure in the spectra are signals for E(1530)0 and EE(1530)0 in neutral channels. Fig. 1f gives sum of invariant mass distributions of all four charge channels after subtraction of background. Particularly, there is no enhancement in mass region around around 1862 MeV/c2, where NA49 observed E3/2 candidates. We determined UL(95%) on Br ■ da/dy|y=o, which at mass of 1862 MeV/c2 are 2.5, 2.3, 0.85 and 3.1 ^b/nucleon in E-n-, E-n+, EE+n+ and EE+n- channels, respectively. The corresponding upper limits using all targets are 2.7,3.2,0.94 and 3.1 n-b /nu-cleon. We also found the UL(95%) on particle ratio Br ■ E—/E0 (1530) < 4% and Br ■ E /E- < 3%. As an illustration, the UL(95%) on nuclear cross section is presented in Fig. 1g (full line) as function of E mass. To conclude, we searched for pentaquark signals in channels p K^ and En. Having found no evidence for signals we set upper limits on production cross sections and particle ratios in mid-rapidity region. If existent, strange pentaquarks (0+ and E3/2 ) also seem to have exotic production mechanisms. References 1. T. Nakano et al. (LEPS Collaboration), Phys. Rev. Lett. 91, 012002 (2003), [arXiv:hep-ex/0301020]. 2. C. Alt et al. (NA49 Collaboration), Phys. Rev. Lett. 92, 042003 (2004), [arXiv:hep-ex/0310014]. 3. I. Abt et al. (HERA-B Collaboration), submitted to Phys. Rev. Lett. (2004), [arXiv:hep-ex/0408048]. Blejske Delavnice Iz Fizike, Letnik 5, St. 1, ISSN 1580-4992 Bled Workshops in Physics, Vol. 5, No. 1 Zbornik delavnice 'Quark Dynamics', Bled, 12. - 19. julij 2004 Proceedings of the Mini-Workshop 'Quark Dynamics', Bled, July 12-19,2004 Uredili in oblikovali Bojan Golli, Mitja Rosina, Simon Sirca Publikacijo sofinancira Ministrstvo za solstvo, znanost in sport Tehnični urednik Vladimir Bensa ZaloZilo: DMFA - zaloZnistvo, Jadranska 19,1000 Ljubljana, Slovenija Natisnila Tiskarna MIGRAF v nakladi 100 izvodov Publikacija DMFA stevilka 1586