Blejske delavnice iz fizike Letnik 18, št. 1 Bled Workshops in Physics Vol. 18, No. 1 ISSN 1580-4992 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2-9, 2017 Edited by Bojan Golli Mitja Rosina Simon Sirca University of Ljubljana and Jozef Stefan Institute dmfa - zaloZništvo Ljubljana, november 2017 The Mini-Workshop Advances in Hadronic Resonances was organized by Society of Mathematicians, Physicists and Astronomers of Slovenia Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana and sponsored by Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Jožef Stefan Institute, Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Organizing Committee Mitja Rosina, Bojan Golli, Simon Sirca List of participants Marko Bračko, Ljubljana, marko.bracko@ijs.si Bojan Golli, Ljubljana, bojan.golli@ijs.si Viktor Kashevarov, Mainz, kashev@kph.uni-mainz.de Itaru Nakagawa, RIKEN, itaru@bnl.gov Hedim Osmanovic, Tuzla, hedim.osmanovic@untz.ba Willi Plessas, Graz, willibald.plessas@uni-graz.at Saša Prelovšek, Ljubljana, sasa.prelovsek@ijs.si Nikita Reichelt, Graz, nikita.reichelt@uni-graz.at Mitja Rosina, Ljubljana, mitja.rosina@ijs.si Helios Sanchis Alepuz, Graz, helios.sanchis-alepuz@uni-graz.at Andrey Sarantsev, Bonn & Gatchina, andsar@hiskp.uni-bonn.de Wolfgang Schweiger, Graz, wolfgang.schweiger@uni-graz.at Simon Sirca, Ljubljana, simon.sirca@fmf.uni-lj.si Jugoslav Stahov, Tuzla, jugoslav.stahov@untz.ba Igor Strakovsky, Washington, igor@gwu.edu Yasuyuki Suzuki, Niigata, yasuyuki_suzuki@riken.jp Alfred Švarc, Zagreb, svarc@irb.hr Lothar Tiator, Mainz, tiator@kph.uni-mainz.de Yannick Wunderlich, Bonn, wunderlich@hiskp.uni-bonn.de Electronic edition http://www-f1.ijs.si/BledPub/ Contents Preface............................................................. V Predgovor..........................................................VII n and n ' photoproduction with EtaMAID including Regge phenomenology V. L. Kashevarov, L. Tiator, M. Ostrick.................................. 1 The role of nucleon resonance via Primakoff effect in the very forward neutron asymmetry in high energy polarized proton-nucleus collision I. Nakagawa........................................................ 6 Single energy partial wave analyses on eta photoproduction - pseudo data H. Osmanovic et al................................................... 17 Cluster Separability in Relativistic Few Body Problems N. Reichelt, W. Schweiger, and W.H. Klink.............................. 22 Baryon Masses and Structures Beyond Valence-Quark Configurations R.A. Schmidt, W. Plessas, and W. Schweiger............................. 30 Single energy partial wave analyses on eta photoproduction - experimental data J. Stahov etal........................................................ 35 Exclusive pion photoproduction on bound neutrons I. Strakovsky........................................................ 39 Resonances and strength functions of few-body systems Y. Suzuki........................................................... 40 From Experimental Data to Pole Parameters in a Model Independent Way (Angle Dependent Continuum Ambiguity and Laurent + Pietarinen Expansion) Alfred Svarc ........................................................ 43 Baryon transition form factors from space-like into time-like regions L. Tiator............................................................ 52 Mathematical aspects of phase rotation ambiguities in partial wave analyses Y. Wunderlich....................................................... 57 Recent Belle Results on Hadron Spectroscopy M. Bracko........................................................... 68 The Roper resonance - a genuine three quark or a dynamically generated resonance? B.Golli............................................................. 76 Possibilities of detecting the DD* dimesons at Belle2 MitjaRosina ........................................................ 82 The study of the Roper resonance in double-polarized pion electropro-duction S. Sirca............................................................. 87 Povzetki v slovenscini............................................... 89 Preface Resonances remain an important tool to study the structure and dynamics of hadrons and an efficient catalyst for our traditional Mini-Workshops at Bled. The many ideas, questions and responses presented at our meeting should not fade away and we thank the participants for submitting their contributions to the Proceedings as a permanent reminder of our common interests and discussions. An important aspect of the talks was the bridge between the phenomenological phase shift analyses and the theoretical interpretation of resonances. Attempts were shown to relate experimental data to pole parameters in a model-independent way, introducing additional constraints to obtain a unique solution. This year, the emphasis was on meson photoproduction, in particular of n and n', as well as doubly-polarized pion electroproduction. Of interest was also pion photoproduction on bound neutrons and the forward neutron asymmetry in proton-nucleus collisions. The Roper resonance is still a challenge. It is not clear to which extent it is predominantly a three-quark system or a dynamically generated resonance. The dynamics of other baryons also requires an extension beyond the valence quark configurations. The knowledge of the baryon form-factors has improved both due to new experimental analyses as well due to new theoretical perspectives, especially regarding transition form-factors. It was interesting to learn about the cluster separability in relativistic few body problems, about phase rotation ambiguities, and about the progress in understanding strength functions in hadronic and nuclear dynamics. The third emphasis was on new resonances in the charm sector. The meson and baryon resonances discovered at the Belle detector at KEKB are still being analysed in order to determine their quantum numbers and their double-qq or "molecular" dimeson structure. In view of the forthcoming Belle2 upgrade it is time to analyse the prospects of identifying the double charm baryons and the DD* dimesons (tetraquarks). We were very happy to host such enthusiastic participants. We do hope to meet you at Bled again soon and that you will enjoy reading these Proceedings and refresh your memories of the subjects of our common interest. Perhaps you might wish to offer these Proceedings to your colleagues as a temptation to join us at Bled in the near future. Ljubljana, November 2017 B. Golli, M. Rosina, S. Sirca The full color version of the Proceedings are available at http://www-f1.ijs.si/BledPub, and the presentations can be found at http://www-f1.ijs.si/Bled2 017/Program.html. Predgovor Resonance so se vedno pomembno orodje za študijzgradbe in dinamike hadro-nov, pa tudi učinkovit katalizator za naše tradicionalne Blejske delavnice. Mnoge zamisli, vprasanja in odzivi, ki smo jih predstavili na nasem srečanju, ne smejo oveneti, zato se zahvaljujemo udelezencem, da so poslali svoje prispevke kot trajen spomin na nasa skupna zanimanja in razprave. Pomemben vidik predavanjje bil most med fenomenolosko analizo faznih premikov in teoreticnim tolmacenjen resonanc. Predstavljeni so bili poskusi, kako povezati eksperimentalne podatke s parametri polov na modelsko neodvisen nacin, s tem da se vpeljejo dodatni pogoji, ki vodijo do enolicne resitve. Letos je bil poudarek na fotoprodukciji mezonov, zlasti n in n', pa tudi na elektropro-dukciji dvojno polariziranih pionov. Zanimiva je bila tudi fotoprodukcija pionov na vezanih nevtronih ter asimetrija nevtronov, ki letijo naprej pri trkih protonov na jedrih. Roperjeva resonanca predstavlja se vedno izziv. Ni se jasno, do katere mere je pretezno sistem treh kvarkov ali dinamicno povzrocena resonanca. Dinamika mnogih drugih barionov tudi zahteva razsiritev modelov na konfiguracije, ki presegajo zgoljvalencne kvarke. Poznavanje barionskih oblikovnih faktorjev se je izpopolnilo zaradi novih eksperimentalnih analiz kakor tudi zaradi novih teo-reticnih pogledov, zlasti v zvezi z oblikovnimi faktorji za prehode. Zanimiv je bil vpogled v locljivost gruc pri relativisticnem problemu malo teles, v mnogolicnost rotacije faz, pa tudi napredek pri razumevanju jakostnih funkcij v hadronski in jedrski dinamiki. Tretji poudarek je bil na novih resonancah v carobnem sektorju. Mezonske in bar-ionske resonance, ki so jih odkrili na detektorju Belle na pospesevalniku KEKB, se vedno analizirajo, da bi dolocili njihova kvantna stevila in njihovo "molekularno" dimezonsko zgradbo v zvezi z dvojnimi pari qq. V perspektivi skorajšnjega pove-canja detektorja Belle2 je cas, da prevetrimo moznosti identifikacije dvojno carob-nih barionov ter dimezonov (tetrakvarkov) DD*. Čutimo se srecne, da smo se druzili s tako navdusenimi udeleženci. Upamo, da vas bomo spet kmalu videli na Bledu in da boste uzivali branje tega Zbornika in osvezili spomine na probleme nasega skupnega zanimanja. Morda boste ponudili ta Zbornik svojim kolegom kot vabo, da se nam v bližnji prihodnosti pridružijo na Bledu. Ljubljana, november 2017 B. Golli, M. Rosina, S. Sirca Barvno verzijo lahko dobite na http://www-f1.ijs.si/BledPub in prosoj-nice predavanjna http://www-f1.ijs.si/Bled2017/Program.html. 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L. Kashevarov, L. Tiator, M. Ostrick Institut fiir Kernphysik, Johannes Gutenberg-Universitat, D-55099 Mainz, Germany Abstract. We present a new version of the EtaMAID model for n and n' photoproduction on nucleons. The model includes 23 nucleon resonances parameterized with Breit-Wigner shapes. The background is described by vector and axial-vector meson exchanges in the t channel using the Regge cut phenomenology. Parameters of the resonances were obtained from a fit to available experimental data for n and n' photoproduction on protons and neutrons. The nature of the most interesting observations in the data is discussed. EtaMAID is an isobar model [1,2] for n and n' photo- and electroproduction on nucleons. The model includes a non-resonant background, which consists of nucleon Born terms in the s and u channels and the vector meson exchange in the t channel, and s-channel resonance excitations, parameterized by Breit-Wigner functions with energy dependent widths. The EtaMAID-2003 version describes the experimental data available in 2002 reasonably well, but fails to reproduce the newer polarization data obtained in Mainz [3]. During the last two years the EtaMAID model was updated [4-6] to describe the new data for n and n' photoproduction on the proton. The presented EtaMAID version includes also n and n' photoproduction on the neutron. At high energies, W > 3 GeV, Regge cut phenomenology was applied. The models include t-channel exchanges of vector (p and and axial vector (bi and hi) mesons as Regge trajectories. In addition to the Regge trajectories, also Regge cuts from rescattering pp, pf2 and ^p, were added, where p is the Pomeron with quantum numbers of the vacuum 0+ (0++) and f2 is a tensor meson with quantum numbers 0+(2++). The obtained solution describes the data up to EY = 8 GeV very well. For more details see Ref. [7]. Energies below W = 2.5 GeV are dominated by nucleon resonances in the s channel. All known resonances with an overall rating of two stars and more were included in the fit. To avoid double counting from s and t channels in the resonance region, low partial waves with L up to 4 were subtracted from the t-channel Regge contribution. The most interesting fit results are presented in Figs. 1-5 together with corresponding experimental data. In Fig. 1, the total yp —» np cross section is shown. A key role in the description of the investigated reactions is played by three s-wave resonances N(1535) 1/2-, * Talk presented by V. Kashevarov 10 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 W [GeV] Fig. 1. (Color online) Total cross section of the yp —> np reaction with partial contributions of the main nucleon resonances. Red line: New EtaMAID solution. Vertical lines correspond to thresholds of KI and n 'N photoproduction. Data: A2MAMI-17 [6]. Fig. 2. (Color online) Total cross section of the yp —> n 'p reaction with partial contributions of the main nucleon resonances. Red line: New EtaMAID solution. Data: A2MAMI-17 [6], CBELSA/TAPS-09 [9], and CLAS-09 [10]. N(1650) 1/2-, and N(1895) 1/2-, see partial contributions of these resonances in Fig. 1. The first two give the main contribution to the total cross section and are known very well. An interference of these two resonances is mainly responsible for the dip at W = 1.68 GeV. However, the narrowness of this dip we explain as a threshold effect due to the opening of the KI decay channel of the N (1650) 1/2-resonance. The third one, N(1895)1/2-, has only a 2-star overall status according to the PDG review [10]. But we have found that namely this resonance is responsible for the cusp effect at W = 1.96 GeV (see magenta line in Fig. 1) and provides a fast increase of the total cross section in the yp —» n 'p reaction near threshold (see black line in Fig. 2). A good agreement with the experimental data was obtained for the cross sections of the YP —» n 'p reaction, Fig. 2. The main contributions to this reaction come from N(1895) 1/2-, N (1900)3/2+, and N(2130)3/2-resonances. W [GeV] Fig. 3. (Color online) Total cross section of the yu —> nu reaction with partial contributions of the main nucleon resonances. Red line: New EtaMAID solution. Data: A2MAMI-14 [11]. Very interesting results were obtained during the last few years for the yn —» ■qu reaction. The excitation function for this reaction shows an unexpected narrow structure at W ~ 1.68 GeV, which is not observed in yp —» np. As an example, the total cross section measured with highest statistics in Mainz [11] is shown in Fig. 3. The nature of the narrow structure has been explained by different authors as a new exotic nucleon resonance, or a contribution of intermediate strangeness loops, or interference effects of known nucleon resonances, see Ref. [12]. In our analyses, the narrow structure is explained as the interference of s, p, and d waves, see partial contributions of the resonances in Fig. 3. Our full solution, red line in Fig. 3, describes the data up to W ~ 1.85 GeV reasonably well and shows a cusp-like structure at W = 1.896 GeV similar as in Fig. 2 for the Yp —» np reaction. However, the data demonstrate a cusp-like effect at the energy of ~ 50 MeV below. This remains an open question for our analyses as well as for the final state effects in the data analysis. Recently, the CLAS collaboration reported a measurement of the beam asymmetry I for both yp —» np and yp —> n'p reactions [13]. At high energies, W > 2 GeV, the yp —> np data have maximal I asymmetry at forward and backward directions, see Fig. 4. We have found that an interference of N(2120)3/2-and N(2060)5/2- resonances is responsible for such an angular dependence. The data was refitted excluding the resonances with mass around 2 GeV. The most significant effect we have found by refitting without N(2120)3/2- (black line) and N(2060)5/2- (blue line). The red line is our full solution. The beam asymmetry I for yp —> n 'p reaction is presented in Fig. 5 with the GRAAL data [14] having a nodal structure near threshold. Such a shape of the an- ■0.51. W=1975 MeV -1 0 1-1 0 1-1 0 1-1 0 2031 J 2055 0 1-1 0 1-1 0 1 0QS©n Fig. 4. (Color online) Beam asymmetry Z for the yp —;> np reaction. Red line: New EtaMAID solution. Results of the refit to the data without N(2120)3/2- are shown by the black lines and without N(2060)5/2- - blue lines. Data: CLAS-17 [13], 0.5 0.5 W=1.9032 GeV 2.006 1.9125 1-1 1.93 1-1 h i 1-1 0 cos 0 Fig. 5. Beam asymmetry Z for the yp —> n 'p reaction. Red line: New EtaMAID solution. Data: GRAAL-15 [14] (black), CLAS-17 [13] (red). 1 -1 0 0 0 0 0 gular dependence could be explained by interference of s and f or p and d waves. However, the energy dependence is inverted in all models. The EtaMAID-2016 solution [5] describes the shape of the GRAAL data for I, but not the magnitude. The new CLAS data [13] can not solve this problem because of poor statistics new threshold. Our new solution describes the I data well at W > 1.95 GeV. In summary, we have presented a new version nMAID-2017n updated with new resonances and new experimental data. The model describes the data currently available for both n and n' photoproduction on protons and neutrons. The cusp in the n total cross section, in connection with the steep rise of the n' total cross section from its threshold, is explained by a strong coupling of the N(1895)1/2- to both channels. The narrow bump in nn and the dip in np channels have a different origin: the first is a result of an interference of a few resonances, and the second is a threshold effect due to the opening of the KI decay channel of the N (1650) 1/2- resonance. The angular dependence of I for yp —» np at W > 2 GeV is explained by an interference of N(2120)3/2- and N(2060)5/2-resonances. The near threshold behavior of I for yp —» n 'p, as seen in the GRAAL data, is still an open question. A further improvement of our analysis will be possible with additional polarization observables which soon should come from the A2MAMI, CBELSA/TAPS, and CLAS collaborations. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 1044). References 1. W. -T. Chiang, S. N. Yang, L. Tiator, and D. Drechsel, Nucl. Phys. A700, 429 (2002). 2. W.-T. Chiang, S. N. Yang, L. Tiator, M. Vanderhaeghen, and D. Drechsel, Phys. Rev. C 68, 045202 (2003). 3. J. Akondi et al. (A2 Collaboration at MAMI), Phys. Rev. Lett. 113,102001 (2014). 4. V. L. Kashevarov, M. Ostrick, L. Tiator, Bled Workshops in Physics, Vol.16, No.1, 9 (2015). 5. V. L. Kashevarov, M. Ostrick, L. Tiator, JPS Conf. Proc. 13, 020029, (2017). 6. V. L. Kashevarov et al. (A2 Collaboration at MAMI), Phys. Rev. Lett. 118, 212001 (2017). 7. V. L. Kashevarov, M. Ostrick, L. Tiator, Phys. Rev. C 96 035207 (2017). 8. C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40,100001 (2016). 9. V. Crede et al. (CBELSA/TAPS Collaboration), Phys. Rev. C 80, 055202 (2009). 10. M. Williams et al. (CLAS Collaboration), Phys. Rev. C 80, 045213 (2009). 11. (A2 Collaboration at MAMI), D. Werthmüller et al., Phys. Rev. C 90, 015205 (2014). 12. (A2 Collaboration at MAMI), L. Witthauer et al., Phys. Rev. C 95, 055201 (2017). 13. P. Collins et al, (CLAS Collaboration), Phys. Lett. B 771, 213 (2017). 14. P. Levi Sandri et al. (GRAAL Collaboration), Eur. Phys. J. A 51, 77 (2015). Bled Workshops in Physics Vol. 18, No. 1 P. 6 Bled, Slovenia, July 2 - 9, 2017 The role of nucleon resonance via Primakoff effect in the very forward neutron asymmetry in high energy polarized proton-nucleus collision I. Nakagawa for the PHENIX Collaboration RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Abstract. A strikingly strong atomic mass dependence was discovered in the single spin asymmetry of the very forward neutron production in transversely polarized proton-nucleus collision at y/s = 200 GeV in PHENIX experiment at RHIC. Such a drastic dependence was far beyond expectation from conventional hadronic interaction models. A theoretical attempt is made to explain the A-dependence within the framework of the ultra peripheral collision (Primakoff) effect in this document using the Mainz unitary isobar (MAID2007) model to estimate the asymmetry. The resulting calculation well reproduced the neutron asymmetry data in combination of the asymmetry comes from hadronic amplitudes. The present EM interaction calculation is confirmed to give consistent picture with the existing asymmetry results in pT + Pb —> n0 + p + Pb at Fermi lab. 1 Nuclear Dependence of Spin Asymmetry of Forward Neutron Production Large single spin asymmetries in very forward neutron production seen [1] using the PHENIX zero-degree calorimeters [2] are a long established feature of transversely polarized proton-proton collisions at RHIC in collision energy y/s = 200 GeV. Neutron production near zero degrees is well described by the one-pion exchange (OPE) framework. The absorptive correction to the OPE generates the asymmetry as a consequence of a phase shift between the spin flip and non-spin flip amplitudes. However, the amplitude predicted by the OPE is too small to explain the large observed asymmetries. A model introducing interference of pion and a -Reggeon exchanges has been successful in reproducing the experimental data [3]. The forward neutron asymmetry is formulated as where ^flip (^non-flip) is spin flip (spin non-flip) amplitude between incident proton and out-going neutron, and 6 is the relative phase between these two amplitudes. Although the OPE can contribute to both spin flip and non-flip amplitudes, resulting AN is small due to the small relative phase. The decent amplitude can be generated only by introducing the interference between spin flip n exchange and spin non-flip a -Reggeon exchange which has large phase shift in between [3]. An 0.5 1 0.3 n is calculated using SOPHIA model [9]. The neutron energy cut xF = En/Ep > 0.4 is applied to be consistent with the experiment [4] where En is the energy of the outgoing neutron and Ep is the incident proton beam energy. As can be seen, the prominent peak is located slightly below A(1232MeV) peak since the equivalent photon yield is weighted to lower energy in the nuclear Coulomb field [10]. The momentum transfer are defined t' = t — (W2 — m2)2/4P2 for the Fermi experiment1, whereas t is defined as —t = m^(1 — xF)2/xF + /xF for the RHIC experiment, where mn is neutron mass, and Pt is the transverse momentum of neutron. Unfortunately, the momentum transfers are not defined consistently between two experiments due to undetected tt+ in the RHIC experiment. -t GeV2 Fig. 4. (Top) The t distributions of the n0p system in pT + Pb n0 + p + Pb for W < 1.36 GeV and 1.36 < W < 1.52 GeV, respectively. The finite asymmetry was observed in the region |t' | < 1 x 10-3 (GeV/c)2 of panel (b) [7]. (Bottom) The experimental momentum transfer distributions of the RHIC experiment for 3 different trigger selections. (Color online) 1 See reference [7] for the definition. Table 1. The difference of experimental conditions between RHIC [4] and Fermi [7] experiments. Fermi RHIC Beam Energy Ep [GeV] 185 100 V [GeV] 19.5 200 Target nucleus Pb Au Detected particle(s) p + n0 n Momentum transfer (GeV/c)2 | t' I < 0.001 0.02 < —t < 0.5 Invariant mass W [GeV] 1.36 < W< 1.52 An —0.57 ± (0.12)sta + 0.21 — 0.18 +0.27 ± 0.003 4 Asymmetry Induced by Photo-Pion Production Pion production reaction from nucleon are intensely studied in various medium energy real photon and electron beam facilities. See reference [11] as one of review articles. The present forward neutron asymmetries via UPC effect corresponds to the photo-pion production from a transversely polarized fixed target. The polarized y*p cross section is given as Eq. (4): n = JqL{Roo + Py R0y} (3) Iql [R0o{1 + P2 cos ^„T (e;)}] (4) where R00 is the unpolarized, while R°y is target polarized response functions, respectively. T(en) corresponds to the definition of the present analyzing power An = T(en) = R°y/R?°. en represents production angle of n in the center-of-mass system. There are several theoretical/phenomenological fitting models available to describe photo-pion production observables. Here I quote Mainz unitary isobar model, namely MAID2007 [12] to calculate the asymmetries in the present kinematics. Shown in Fig. 5 is the MAID prediction of the unpolarized response function R0o plotted as a function of the invariant mass W of pion and nucleon systems at Q2 = 0(GeV/c)2 and en = 40°. The multipoles are weak function of Q2(= —t) and only moderately change within our kinematic coverage —t < 0.5 (GeV/c)2. The leading order multipole decomposition following the notation of reference [13] is given in Eq. (5): R?° = 5I Mi + I 2 + MÎ+MT- + 3M|+E1+ + ... (5) 2 where Mi+ is famous magnetic dipole transition amplitude from the nucleon ground state to the A(P33) resonance state. As blue curve indicates, the y*p —» LUy* 25 20 15 g 10 t£ R00 (Q2=0, theta*pi=40deg) 1100 1200 1300 1400 Invariant Mass [MeV] 1500 n + pi + p + pi0 5 0 Fig. 5. (Color online) Unpolarized R0O(W) response function at Q2 = 0(GeV/c)2 and = 40° plotted as a function of the invariant mass W [MeV]. Red and blue curves represent MAID predictions for y*p —> + n and y*p —> n0 + p decay channels, respectively. n0p channel shows distinctive peak around well known A resonance region (W = 1232 MeV) in Fig. 5. This is mainly driven by the dominant |M1+|2 term in Eq. (5). On the contrary, the A peak is not as distinctive as n0 channel for the channel and shows rather larger cross section in the threshold pion production region below A. This is due to enhanced charge coupling of photon to the pion field in the target proton which doesn't exist for n0 channel. This is known as KrollRudermann term [14] as shown in the diagram (d) in Fig. 3. Shown in Fig. 6 is the target polarization response function R°y (W) of the MAID predictions for —» n+n (red) and y*p^ —» n0p (blue) decay channels, respectively. The leading order multipole decomposition of RTy is denoted as Eq. (6): R0y = Im{E0+(E1+ - M1 + )-4cos0n(E1+M1 + )....} (6) The asymmetries show peak structure around A region for both and n0 channels, while the sign is opposite. The magnitude of asymmetry is substantially as large as RTy - 15|p.b/st] for channel compared to n0 channel. This is because of the strong interference between E0+ and M1+ channel in channel as appears in the first term in Eqn.6. The amplitude of E0+ is much greater in channel compared to n0 channel due to aforementioned Kroll-Rudermann term. Although dominant A amplitude, i.e. M1+ is even stronger in n0 channel, this interference is relatively minor due to smallness of E0+ for n0 channel. The obtained analyzing power AN for MAID predictions by taking the ratio of response functions RTy (W) and R00 (W) are shown in Fig. 7 plotted as a function of the invariant mass W at Q2 = 0(GeV/c)2 and 0^ = 40°. Note there are distinctive difference between and n0 channels in AN as a function of W according to the MAID model. shows remarkably large asymmetry over AN > 0.8 R0y (Q2=0, theta*pi=40deg) 20 15 10 5 2 o -5 -10 n + pi + p + piO 15OO Invariant Mass [MeV] Fig. 6. (Color online) Polarized R0y (W) response function at Q2 = 0(GeV/c)2 and = 40° plotted as a function of the invariant mass W [MeV]. Red and blue curves represent MAID predictions for y*pT —> n+n and y*pT —> n0p decay channels, respectively. just below A(1232 MeV) due to the interference between E0+ of Kroll-Rudermann and A dipole resonance Mi+ terms. The contribution of this invariant mass region to the observed neutron is large due to matching peak of the invariant mass yield as shown in the right panel of Fig.2. Analyzing Power (Q2=0, theta*pi=40deg) Fig. 7. (Color online) Analyzing power AN(W) at Q2 = 0(GeV/c)2 and 6^ = 40° plotted as a function of the invariant mass W [MeV]. Red and blue curves represent MAID [12] predictions for y*pT —> n+n and y*pT —> n0p decay channels, respectively. The MAID is in general known to fit reasonably well to photo-pion production data in low to medium energy region. Shown in Fig. 8 is the analyzing power T(= AN) of MAID (red curve) fits to —> n+p reaction data observed in PHOENICS experiment at ELSA [15]. For the comparison, Argonne-Osaka [16] model fits are also shown in blue curve. Although some model dependence is seen in higher energies W > 1365 MeV in the region where no data, two models are fairly consistent to each other in lower energies W < 1319 MeV. Although the ELSA data is not necessarily perfect overlap with the kinematic range of the present RHIC data, the extrapolation of data by MAID seem to give reasonable estimate since the data coverage is sufficiently large in W bins below A which are rather weighted for the present neutron data. - - 5: : 5 ' ■ 5. - 05/TSs^n 'E, = 220 MeV ' -W = 1137 MeV 0 50 100 150 5 'E, = 241 MeV " - W = 1154 MeV 0 50 100 l50 0 5'E, = 262 MeV ' -W = 1171 MeV 0 50 100 l.0 0 5'E, = 282 MeV ' -W = 1187 MeV 0 50 100 l.0 5 'E, = 303 MeV ' - W = 1203 MeV 0 50 100 l.0 05 'E, = 324 MeV ■ W = 1219 MeV 0 50 100 l.0 'E, = 345 MeV ' -W = 1236 MeV 0 50 100 150 5 'E, = 366 MeV " " W = 1251 MeV 0 50 100 l.0 0 5 "E, = 393 MeV ' -W = 1271 MeV 0 50 100 l.0 0 5 "E, = 425 MeV ' -W = 1295 MeV 0 50 100 l.0 5"E, = 458 MeV " - W = 1319 MeV 0 50 100 l.0 05"E, = 49l MeV ' W = 1342 MeV 0 50 100 l.0 ■ . 'E, = 524 MeV ' -W = 1365 MeV 5 'E, = 557 MeV ' -W = 1387 MeV 0 5 "E, = 589 MeV ' W = 1409 MeV 0.5 • I • 0- ^ - ^ 0 5'E, = 620 MeV ' -W = 1429 MeV 5'E, = 650 MeV ' -W = 1449 MeV W = 1466 MeV s, s, 0, 0, 0, 0, Fig. 8. (Color online) Analyzing power T(= AN) of MAID (red curve) and Argonne-Osaka [16] model (blue curve) fit to y*pT —> n+p reaction data observed in PHOENICS experiment at ELSA [15]. In reference [17], an attempt is made to evaluate average AN within the present RHIC experiment using so evaluated MAID An. Shown in the left panel of the Fig.9 is the analyzing power T(= AN) as a function of pion production angle 0^ and the invariant mass W of —> n. The region between thin and thick curves are the rapidity range of the present RHIC experiment and each curves corresponds to the rapidity boundaries of n = 8.0 and n = 6.8, respectively. As can be seen in the figure, the large AN >0.8 is distributed in 0^ <1 [rad] around W ~ 1.2 GeV and this is where the peak of the neutron yield is located as shown in the right panel of Fig.2 according to EM interaction Monte-Carlo. The yield weighted average of AN within the acceptance between 6.8 < n < 8.0 and xF > 0.4 is plotted as open square in the right panel of Fig.9. The analyzing power via EM interaction are very similar between p+Al or p+Au because the slope of the photon yield as a function of photon energy is very similar. On the other hand, resulting AN will be quite different between them due to the fraction of hadronic interaction and the EM interactions are quite different. In fact, the EM cross section grows square function of atomic number Z. The fraction of the hadronic and EM interactions are estimated by the cross section ratio of them assuming one pion exchange (OPE) for the hadronic interaction. The is simpler hadronic interaction model than the reference [5]. However, the cross section of the hadronic interaction for the leading neutron production in this very forward rapidity range 6.8 < n < 8.0 is known to be dominated by OPE [3]. On the other hand, the nuclear absorption effect is claimed to play important role in the reference [5] and is not considered in reference [17] though, the absorption effects are somewhat canceled when one take ratio between the hadronic and the EM interactions. Details are discussed in the reference [17]. So obtained hadron/EM cross section weighted AN are plotted as open circles in the right panel of Fig.9 and are compared with experimental analyzing power data (solid symbols). Solid circle and squares are inclusive and BBC vetoed data, respectively. The calculated AN open circles are to be compared with inclusive data points (solid circle) and they are in very good agreement. 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0.4 0.2 1.2 1.4 1.6 1. W (GeV) □ □ ■ Simulation (UPC) Simulation (UPC + Hadronic) q —•— PHENIX (pA, inclusive) - —■— PHENIX (pA, veto) —A— PHENIX (pp) 50 100~ Atomic number Z 3 2 1 0 0 0 Fig.9. (left) Analyzing power T(= An) as a function of pion production angle in and the invariant mass W of y*pT —> n+n. The region between thin and thick curves are the rapidity range of the present RHIC experiment and each curves corresponds to the rapidity boundaries of n = 8.0 and n = 6.8, respectively. (right) Comparison of experimental analyzing power data (solid symbols) and model predictions (open symbols) plotted as a function of atomic number Z. Solid circle and squares are inclusive and BBC vetoed data, respectively. Open square is kinematically averaged AN prediction over RHIC acceptance by MAID. Open circles are weighted mean prediction of MAID and one pion exchange AN for Al and Au. Both plots are quoted from reference [17]. 5 Summary A theoretical attempt was made to explain strong A-dependence in the very forward neutron asymmetry recently observed in transversely polarized proton-nucleus collision at a/s=200 GeV in PHENIX experiment at RHIC [4]. The drastic A-dependence in the forward neutron asymmetry AN cannot be explained by the conventional hadronic interaction model [5] which was successful to explain the asymmetries observed for p + p collision [3]. In this document, possible major contribution in the asymmetry from the UPC (Primakoff) effect via one photon exchange from the nuclear Coulomb field is discussed. The Mainz unitary isobar (MAID2007) model [12] was used to estimate the asymmetry by the EM interaction which fit past y * pT —> n+n reaction data [15] well. The MAID predicts large asymmetry below A region for n+n-channel due to the interference between non-resonance contact Eo+ (non-spin flip) and A resonance Mi+ (spin flip) amplitudes. Once kinematic average within the detector acceptance and kinematic cuts, the resulting asymmetries overshot both inclusive AN data for both p + Al and p + Au data. Once these average EM asymmetries are further taken weighted mean by cross section ratio with hadronic asymmetries, the resulting asymmetries reproduced both p + Al and p + Au data well [17]. The importance of the interference in non-resonance and A resonance contradicts from the large asymmetry observed in pT + Pb —» n0 + p + Pb at Fermi lab [7] which is interpreted mainly due to the interference between A and N*(1440) and higher resonances. This difference can be explained by the relatively strong Kroll-Rudermann term [14] contribution for n+ channel, and which raises the importance of the interference below A unlike n0 channel. The present EM asymmetry calculation framework is confirmed to be at least qualitatively consistent with the claim made by the authors of Fermi experiment [7]. References 1. Y. Fukao et al., Phys. Lett. B 650, 325 (2007), A. Adare et al., Phys. Rev. D 88, 032006 (2013). 2. C. Adler, A. Denisov, E. Garcia, M. J. Murray, H. Strobele, and S. N. White, Nucl. Instrum. Methods Phys. Res., A 470, 488 (2001). 3. B. Z. Kopeliovich, I. K. Potashnikova, and Ivan Schmidt, Phys. Rev. Lett. 64,357 (1990). 4. The PHENIX collaboration, arXiv:1703.10941. 5. B. Z. Kopeliovich, I. K. Potashnikova, and Ivan Schmidt, arXiv:1702.07708. 6. I. G. Alekseev et al.: Phys. Rev. D 79, 094014 (2009). 7. D. C. Carey et al., Phys. Rev. D 79, 094014 (2009). 8. S. R. Klein, J. Nystrand, Phys. Rev. C 60, 014903 (1999), STARLIGHT webpage, http://starlight.hepforge.org/ 9. A. Mucke, R. Engel, J.P. Rachen, R.J. Protheroe, and T. Stanev, Comput. Phys. Commun. 123, 290-314 (2000); SOPHIA webpage, http://homepage.uibk.ac.at/ c705282/S0PHIA.html 10. G. Mitsuka, Eur. Phys. J. C 75:614 (2015). 11. Aron M. Bernstein, Mohammad W. Ahmed, Sean Stave, Ying K. Wu, and Henry R. Weller, Annu. Rev. Nucl. Part. Sci. 2009.59:115-144. 12. D. Drechsel, S. S. Kamalov, and L. Tiator, Unitary isobar model (MAID2007), Eur. Phys. J. A 34, 69 (2007). 13. D. Drechsel and L. Tiator, J. Phys. G: Nucl. Part Phys. 18, 449 (1992). 14. Kroll N, Ruderman, MA. Phys. Rev. 93, 233 (1954). 15. H. Dutz et al., Nucl. Phys. A 601, 319 (1996). 16. H. Kamano, S. X. Nakamura, T. -S. H. Lee, and T. Sato, Phys. Rev. C 88, 035209 (2013). 17. G. Mitsuka, Phys. Rev. C 95, 044908 (2017). Bled Workshops in Physics Vol. 18, No. 1 p. 17 Bled, Slovenia, July 2 - 9, 2017 Single energy partial wave analyses on eta photoproduction - pseudo data H. Osmanovic*a, M. Hadzimehmedovica, R. Omerovica, S. Smajica, J. Stahova, V. Kashevarovb, K. Nikonovb, M. Ostrickb, L. Tiatorb, A. Svarcc aUniversity of Tuzla, Faculty of Natural Sciences and Mathematics, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina bInstitut fur Kernphysik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany cRudjer Boskovic Institute, Bijenicka cesta 54, P.O. Box 180,10002 Zagreb, Croatia Abstract. We perform partial wave analysis of the eta photoproduction data. In an iterative procedure fixed-t amplitude analysis and a conventional single energy partial wave analysis are combined in such a way that output from one analysis is used as a constraint in another. To demonstrate the modus operandi of our method it is applied on a well defined, complete set of pseudo data generated within EtaMAID15 model. 1 Introduction Single energy partial wave analysis (SE PWA) is a standard method used to obtain partial waves from scattering data at a given energy. Invariant amplitudes, reconstructed from partial waves by means of corresponding partial wave expansions obey a fixed-s analyticity required in Mandelstam hypothesis. It is quite general that at a given energy many different partial wave solutions equally well describe the data. The fit to the data at one energy "does not know" which solution was obtained in independent SE PWA at another, even neighboring energies. This poses a problem of finding a unique partial wave solution as a function of energy. To solve this problem and to achieve continuity of partial wave solution in energy, one has to impose some additional constraints on partial wave solutions. The aim of this paper is to demonstrate a method which imposes analyticity of invariant scattering amplitudes at fixed values of Mandelstam variable t in addition to analyticity at fixed s-value which is already achieved by partial wave expansion. In our method SE PWA and a fixed-t amplitude analysis (Ft AA) are coupled together in an iterative procedure in such a way that output from one analysis serves as a constraint in another. Detailed description of formalism and the method is given in refs. [2], [2]. Here we demonstrate how the method works. As an input we use the eta photoproduction pseudo data constructed from theoretical model EtaMAID-2015 [3]. Applying our method, we reproduced partial waves from a model which was used to generate the data fitted. This proves uniqueness of partial wave solution obtained applying our method. * Talk presented by H. Osmanovic 2 Method and results To prove uniqueness of solution obtained by use of our method, we generated a complete set of observables in the eta photoproduction process: {ct0, £, T, P, F, G, C x / ,0 x'} [4,5]. To apply our method we need data at two different kinematical grids: energy -1 (W,t) to be used in the Ft AA, and energy - scattering angle theta grid to be used in SE PWA. Our pseudo data sets are generated at 140 energies inside the physical region, each at 50 t-values with artificially small errors of 0.1%. W-t kinematical grid is shown in Fig. 1. Yellow line shows the data used in the Fig. 1. (Color online) (Wcm, t) diagram for r| photoproduction. Points represent pseudodata generated by EtaMAID2015a model in physical range. Yellow line symbolizes fixed-t analysis, and red line symbolizes fixed-s (SE) analysis. Ft AA ( t = -0.6GeV2), while the data along red line ( W = 1800MeV) are used in the SE PWA. Iterative procedure in our method is shown in Fig. 2. XsEdata and Fig. 2. Iterative procedure in a combined single energy partial wave analysis and fixed-t amplitude analysis. Single energy partial wave analyses on eta photoproduction - pseudo data 19 X2Tdata are standard quadratic forms used in fitting the data, ®conv is convergence test function which is integral part of Pietarinen expansion method used in Ft AA [6-10], while ®trunc makes a soft cut off of higher multipoles at lower energies in SE PWA (for technical details see refs [2], [2]). The two analyses, SE PWA and Ft AA, are coupled by terms x2t and x|E which measure deviations of values of invariant amplitudes obtained in SE PWA from corresponding ones obtained in Ft AA and vice versa. After several iterations, usually not more than three, results from both analyses agree reasonably well. Figure 3 and Figure 4 show importance of constraint from Ft AA in obtaining a unique partial wave solution in SE PWA. In Figure 3 are shown partial waves obtained in unconstrained SE 20 1500 1800 2100 W[MeV] 0.03 0 -0.03 ■ 0.08 ■ 0.04 0 ■g 0.3 1 + 0 LU -0.3 1.5 1500 1800 2100 W[MeV] 1500 1800 2100 1500 1800 2100 ■ 0.15 1500 1800 2100 W[MeV] 1500 1800 2100 W[MeV] 1500 1800 2100 W[MeV] , 0.003 0 -0.003 • „ o •.."» 1500 1800 2100 0.03 IT 0.02 EE "T 0.01 1500 1800 2100 W[MeV] 1 b 0.5 r\l LU 0 0.5 1 0.5 b F 0 r\l M 0.5 1 0.02 b 0 ^t- LU -0.02 0.06 E 0 04 (- 1—' 0.02 ^t- M 0 -0.02 W[MeV] o ° »□ 1500 1800 2100 1500 1800 2100 W[MeV] 0 Fig. 3. (Color online) The result of on unconstrained single-energy fit described in the text. The blue and red points show the real and imaginary parts of the multipoles obtained in the fit compared to the "true" multipoles from the underlying EtaMAID-2015 model (blue and red solid lines). PWA. Even if a complete set of data with small errors is used in analysis, unique solution is not obtained- input partial waves solution from which the data is generated is not reconstructed. Figure 4 shows results of PWA using our method with the same input data after two iterations. Starting solution is reconstructed with a high accuracy. Fig. 4. (Color online) Real (blue) and imaginary (red) parts of electric and magnetic multi-poles up to L = 4. The points are the result of the analytically constrained single-energy fit to the pseudo data and are compared to the multipoles of the underlying EtaMAID-2015 model, shown as solid lines. 3 Conclusions In order to achieve unique and continuous solution in energy, additional constraint in an partial wave analysis is needed. It is shown that a unique solution may be obtained using only analytic properties of invariant scattering amplitudes at fixed values of Mandelstam variables s and t as constraint. References 1. H. Osmanovic, M. Hadzimehmedovic, R. Omerovic, J. Stahov,V. Kashevarov, K. Nikonov, M. Ostrick, L. Tiator, and A. Svarc, arXiv:1707.07891 [nucl-th] 2. M. Hadzimehmedovic, V. Kashevarov, K. Nikonov, R. Omerovic, H. Osmanovic, M. Ostrick, J. Stahov, A. Svarc, L. Tiator, Bled Workshops Phys., 16, 40 (2015). 3. V. L. Kashevarov, L. Tiator, M. Ostrick, Bled Workshops Phys., 16, 9 (2015). 4. Y. Wunderlich, R. Beck and L. Tiator, Phys. Rev. C 89, no. 5, 055203 (2014). 5. W. T. Chiang and F. Tabakin, Phys. Rev. C 55, 2054 (1997). 6. E. Pietarinen, Nuovo Cimento Soc. Ital. Fis. 12A, 522 (1972). 7. E. Pietarinen, Nucl. Phys. B 107 21 (1976). Single energy partial wave analyses on eta photoproduction - pseudo data 21 8. E. Pietarinen, University of Helsinki Preprint, HU-TFT-78-23, (1978). 9. J. Hamilton, J. L. Petersen, New development in Dispersion Theory, Vol. 1. Nordita, Copenhagen, 1973. 10. G. Hohler, Pion Nucleon Scattering, Part 2, Landolt-Bornstein: Elastic and Charge Exchange Scattering of Elementary Particles, Vol. 9b (Springer-Verlag, Berlin, 1983). Bled Workshops in Physics Vol. 18, No. 1 p. 22 Bled, Slovenia, July 2 - 9, 2017 Cluster Separability in Relativistic Few Body Problems* N. Reichel^, W. Schweigera, and W.H. Klinkb institute of Physics, University of Graz, A-8010 Graz, Austria b Department of Physics and Astronomy, University of Iowa, Iowa City, USA b Abstract. A convenient framework for dealing with hadron structure and hadronic physics in the few-GeV energy range is relativistic quantum mechanics. Unlike relativistic quantum field theory, one deals with a fixed, or at least restricted number of degrees of freedom while maintaining relativistic invariance. For systems of interacting particles this is achieved by means of the, so called, "Bakamjian-Thomas construction", which is a systematic procedure for implementing interaction terms in the generators of the Poincare group such that their algebra is preserved. Doing relativistic quantum mechanics in this way one, however, faces a problem connected with the physical requirement of cluster separability as soon as one has more than two interacting particles. Cluster separability, or sometimes also termed "macroscopic causality", is the property that if a system is subdivided into subsystems which are then separated by a sufficiently large spacelike distance, these subsystems should behave independently. In the present contribution we discuss the problem of cluster separability and sketch the procedure to resolve it. 1 Introduction to relativistic quantum mechanics It is a widespread opinion that a relativistically invariant quantum theory of interacting particles has to be a (local) quantum field theory. Therefore we first have to specify what we mean by "relativistic quantum mechanics". Relativistic quantum mechanics is based on a theorem by Bargmann which basically states that [1,2]: A quantum mechanical model formulated on a Hilbert space preserves probabilities in all inertial coordinate systems if and only if the correspondence between states in different inertial coordinate systems can be realized by a single-valued unitary representation of the covering group of the Poincare group. According to this theorem one has succeeded in constructing a relativistically invariant quantum mechanical model, if one has found a representation of the (covering group of the) Poincare group in terms of unitary operators on an appropriate Hilbert space. Equivalently one can also look for a representation of the generators of the Poincare group in terms of self-adjoint operators acting on this Hilbert space. These self-adjoint operators should then satisfy the Poincare * Talk presented by N. Reichelt and by W. Schweiger algebra [JM'^ i eijkJk , [K\Kj] =—ieijkJk , [Ji,Kj]= ieijkKk , [P^,PV] = 0, [Ki,P0]=-iPi, [Ji,P0]= 0, [J\P'] = ieijkPk , [Ki,Pj] =-i6ij P0 . (1) P0 and Pi generate time and space translations, respectively, Ji rotations and Ki Lorentz boosts. From the last commutation relation it is quite obvious that, if P0 contains interactions, Ki or Pj (or both) have to contain interactions too. The form of relativistic dynamics is then characterized by the interaction dependent generators. Dirac [3] identified three prominent forms of relativistic dynamics, the instant form (interactions in P0, Ki, i = 1,2,3), the front form (interactions in P- = P0 — P3, F1 = K1 — J2, F2 = K2 + J1) and the point form (interactions in P^, i = 0,1,2,3). In what follows we will stick to the point form, where P^, the generators of space-time translations, contain interactions and J, K, the generators of Lorentz transformations, are interaction free. The big advantage of this form is that boosts and the addition of angular momenta become simple. For a single free particle and also for several free particles it is quite easy to find Hilbert-space representations of the Poincare generators in terms self-adjoint operators that satisfy the algebra given in Eq. (1), but what about interacting systems? Local quantum field theories provide a relativistic invariant description of interacting systems, but then one has to deal with a complicated many-body theory. It is less known that interacting representations of the Poincare algebra can also be realized on an N-particle Hilbert space and one does not necessarily need a Fock space. A systematic procedure for implementing interactions in the Poincare generators of an N-particle system such that the Poincare algebra is preserved, has been suggest long ago by Bakamjian and Thomas [4]. In the point form this procedure amounts to factorize the four-momentum operator of the interaction-free system into a four-velocity operator and a mass operator and add then interaction terms to the mass operator: P^ = MV£ee = (Mfree + Mintf . (2) Since the mass operator is a Casimir operator of the Poincare group, the constraints on the interaction terms that guarantee Poincare invariance become simply that Mint should be a Lorentz scalar and that it should commute with fee, i.e. [Mint, V£ee] = 0. Remarkably, this kind of construction allows for instantaneous interactions ("interactions at a distance"). Similar procedures can also be carried out in the instant and front forms of relativistic dynamics such that the physical equivalence of all three forms is guaranteed in the sense that the different descriptions are related by unitary transformations [5]. A very convenient basis for representing Bakajian-Thomas (BT) type mass operators consists of velocity states N |v;ki,m;k2,p2;...;^n,Hn) , Y- ki = 0. (3) i=1 These specify the state of an N-particle system by its overall velocity v, the particle momenta k in the rest frame of the system and the spin projections |j.t of the individual particles. The physical momenta of the particles are then given by pt = B(v)kt, where B(v) is a canonical (rotationless) boost with the overall system velocity v. Associated with this kind of boost is also the notion of "canonical spin" which fixes the spin projections N-particle velocity states, as introduced above, are eigenstates of the free N-particle velocity operator fee and the free mass operator Mfree |v;ki, m;k2, m-2; ...) = (^i + ^2 + ...) |v;k, m;k2, ^2;...), (4) with = m? + k?. The overall velocity factors out in velocity-state matrix elements of BT-type mass operators, (v'; kl, m; k2, ^2;... lMlv, ki, m; k2, M2...) co The role of the separation operator will become clearer by means of an example. Let us consider (space-like) separation by a canonical boost. In this case subsystem (A) is boosted with velocity v and subsystem (B) with velocity -v. The action on the wave function is then (uVA)(B)^ (pt€(A), Pj6(B}) = $(b(-v)Pte(A),b{vj?je(B)) (8) and one has to consider the limit ct = |v| —» oo in Eq. (7). Having introduced a separation operator we are now able to formulate cluster separability in a more formal way. In the literature one can find different notions of it. A comparably weak, but physically plausible requirement, is cluster separability of the scattering operator: s-lim Uco It means that the scattering operator should factorize into the scattering operators of the subsystems after separation. For three-particle systems it has been demonstrated that this type of cluster separability can be achieved by a BT construction [6]. A stronger requirement is that the Poincare generators become additive, when the clusters are separated. In a weaker version this means for the four-momentum operator that Jim <®|u = 0, (10) the stronger version is that lim (> - PA) ® I(B) - I(A) ® P^)) UiA)(B)l®) = 0. (11) The BT construction violates both conditions already in the 2+1-body case (i.e. particles 1 and 2 interacting and particle 3 free) [2,7]. The reason for the failure can essentially be traced back in this case to the fact that the BT-type mass operator and the mass operator of the separated 2+1-particle system differ in the velocity conserving delta functions. In the BT-case it is the overall three-particle velocity which is conserved, in the separated case it is rather the velocity of the interacting two-particle system. The separation, however, is done by boosting with the velocity of the interacting two-particle system. One may now ask, whether wrong cluster properties lead to observable physical consequences. From our studies of the electromagnetic structure of mesons we have to conclude that this is indeed the case [8-10]. In these papers electron scattering off a confined quark-antiquark pair was treated within relativistic point form quantum mechanics starting from a BT-type mass operator in which the dynamics of the photon is also fully included. The meson current can then be identified in a unique way from the resulting one-photon-exchange amplitude which has the usual structure, i.e. electron current contracted with the meson current and multiplied with the covariant photon propagator. The covariant analysis of the resulting meson current, however, reveals that it exhibits some unphysical features which most likely can be ascribed to wrong cluster properties. For pseu-doscalar mesons, e.g., its complete covariant decomposition takes on the form f(p'm; pm) = (pm + p'mr f(Q2,s) + (pe + per g(Q2,s). (12) It is still conserved, transforms like a four-vector, but exhibits an unphysical dependence on the electron momenta which manifests itself in form of an additional covariant (and corresponding form factor) and a spurious Mandelstam-s dependence of the form factors. Although unphysical, these features do not spoil the relativistic invariance of the electron-meson scattering amplitude. The Mandelstam-s dependence of the physical and spurious form factors f and g is shown in Fig. 1. Since the spurious form factor g is seen to vanish for large s and the s-dependence of the physical form factor f becomes also negligible in this case it is suggestive to extract the physical form factor in the limit s —» oo. This strategy was pursued in Refs. [8-10] where it lead to sensible results. It gives a simple analytical expression for the physical form factor F(Q2) = lim^^co f(Q2, s) which agrees with corresponding front form calculations in the q^ = 0 frame. Similar effects of wrong cluster properties on electromagnetic form factors were also observed in model calculations done within the framework of front form quantum mechanics [11]. Fig. 1. Mandelstam-s dependence of the physical and spurious B meson electromagnetic form factors f and g for various values of the (negative) squared four-momentum transfer Q2 [9]. The result has been obtained with a harmonic-oscillator wave function with parameters a = 0.55 GeV, mb = 4.8 GeV, mu,d = 0.25 GeV. 3 Restoring cluster separability It is obviously the BT-type structure of the four-momentum operator (see Eq. (2)) which guarantees Poincare invariance on the one hand, but leads to wrong cluster properties on the other hand (if one has more than two particles). In order to show, how this conflict may be resolved, let us consider a three-particle system with pairwise two-particle interactions. To simplify matters we will consider spinless particles and neglect internal quantum numbers. We start with the four-momentum operators of the two-particle subsystems, Pj = M(ij)V(ij), = 1,2,3, i = j, (13) which have a BT-type structure (i.e. Vj is free of interactions). Cluster separability holds for these subsystems, if the two-particle interaction is sufficiently short ranged. The third particle can now be added by means of the usual tensor-product construction p(ij)(k) = P(ij) ® I(k) + I(ij) ® P(k) (14) The individual four-momentum operators P^)^) describe 2+1-body systems in a Poincare invariant way and exhibit also the right cluster properties. One may now think of adding all these four momentum operators, to end up with a four momentum operator for a three particle system with pairwise interactions: p( _ p( i P( i P( _ 2p^ (15) r3 _ r(12)(3) + r(23)(1) + r(31)(2) 2r3 free . (15) But the components of the resulting four-momentum operator do not commute, [P(,PV]_0 since [M(ij) int,V(j]_ 0. (16) One can, of course, write the individual P(lj)(k) in the form P(ij)(k) _ M(ij)(k) lV'(ij)(k) with M^k) _ p(ij)(k) • p(ij)(k), (17) but the four-velocities VjijHk contain interactions and differ for different clusterings, so that an overall four-velocity cannot be factored out of PP3( . The key observation is now that all four-velocity operators have the same spectrum, namely R3. This implies that there exist unitary transformations which relate the four-velocity operators. One can find, in particular, unitary operators U(ij)(k) such that VW) _U(ij)(k)V3lUiij)(k). (18) With these unitary operators one can now define new three-particle momentum operators for a particular clustering, P(ij)(k) U{ij)(k)P(Jlj)(k)U(ij)(k) _ U{ij)(k)M(ij)(k)U(ij)(k)U{ij)(k)1p(ij)(k)U(ij)(k) _ M(ij)(k)V(, (19) which have already BT-structure, i.e. with the free three-particle velocity factored out. From Eq. (19) it can be seen that the unitary operators U(ij)(k) obviously "pack" the interaction dependence of the four-velocity operators V(ij)(k) into the mass operator M(ij)(k). Therefore they were called "packing operators" by Sokolov in his seminal paper on the formal solution of the cluster problem [12]. The sum (P(12)(3) + P(23)(1) + P(?1 )(2) _ 2P(free) describes a three-particle system with pairwise interactions, it has now BT-structure and satisfies thus the correct commutation relation. However, it still violates cluster separability. The solution is a further unitary transformation of the whole sum by means of U _ ^ U(ij)(k), assuming that U(ij)(k) —> 1 for separations (ki)(j), (jk)(i) and (i)(j)(k). The final expression for the three-particle four-momentum operator, that has all the properties it should have, is: p3( :_ U p( + p( + p( + p( _ 2P( r(12)(3) + P(23)(1) + r (31 )(2) + r (123) int 2r3 free U{ _ U [M(12)(3) + M(12)(3) + M(12)(3) + M(123) int _ 2M3 free] V3" U{ _ UM3 V3( U{. (20) If U commutes with Lorentz transformations, it can be shown that such a "generalized BT construction" will satisfy relativity and cluster separability for N-particle systems. In addition to the three-body force induced by U, which is of purely kinematical origin, we have also allowed for a genuine three-body interaction M(i23)int. Since the U(i;j)(k) will, in general, not commute, U depends on the order of the U(tj)(k) in the product. For identical particles one should even take some kind of symmetrized product, for which also different possibilities exist [2,12]. This means that P3 is, apart of the newly introduced free-body interaction M(123) int, not uniquely determined by the two-body momentum operators p3j). There are even different ways to construct the packing operators U(i;j)(k). All the unitary transformations leave, however, the on-shell data (binding energies, scattering phase shifts, etc.) of the two-particle subsystems untouched, they only affect their off-shell behavior. The kind of procedure just outlined formally solves the cluster problem for three-body systems. Generalizations to N > 3 particles and particle production have also been considered [13]. Its practical applicability, however, depends strongly on the capability to calculate the packing operators for a particular system. A possible procedure can also be found in Sokolov's paper. The trick is to split the packing operator further U(ij)(k) = Wt(M(ij) )W(MW) free) (21) into a product of unitary operators which depend on the corresponding two-particle mass operators in a way to be determined. With this splitting one can rewrite Eq. (18) in the form W(M(ij) free)V3lWt(M(ij) free) = W(M(ij)) V^ )(k) W^Mj . (22) Since this equation should hold for any interaction the right- and left-hand sides can be chosen to equal some simple four-velocity operator, for which Vji) <£> Ik is a good choice. In order to compute the action of W it is then convenient to take bases in which matrix elements of V33, Vjij <8> Ik and Vjij)(k) can be calculated. This is the basis of (mixed) velocity eigenstates |V(12); kl, ^2, P3) = |V(!2); kl, £2) ® IP3> (23) of M(ij)(k) free if one wants to calculate the action of W(Mj free) and corresponding eigenstates of M(ij )(k) if one wants to calculate the action of W(M(ij)). It turns out that the effect of these operators is mainly to give the two-particle subsystem (ij) the velocity v(ij)(k) of the whole three-particle system. After some calculations one finds out that the whole effect of the packing operator U(ij)(k) on the mass operator M(ij)(k) is just the replacement /3/2 3/2 v(ij)53(v('ij) - v(ij)) m(3j) m(ij) ^v(ij) • v(ij)(k) ^V(ij) • V(ij)(k) -/3/2--3/2-V(ij)(k)5 (v(ij)(k) - V(ij)(k)) (24) m(ij)(k) m(ij)(k) in the mixed velocity-state matrix elements. Here m(ij) and m(ij )(k) are the invariant masses of the free two-particle subsystem and the free three-particle system, V(ij) and V(ij)(k) the corresponding four-velocities. 4 Summary and outlook We have given a short introduction into the field of relativistic quantum mechanics. It has been shown that the Bakamjian-Thomas construction, the only known systematic procedure to implement interactions such that Poincare invariance of a quantum mechanical system is guaranteed, leads to problems with cluster separability for systems of more than two particles. Cluster separability is a physically sensible requirement for quantum mechanical systems which replaces mi-crocausality in relativistic quantum field theories. We have discussed the physical consequences of wrong cluster properties, e.g., unphysical contributions in electromagnetic currents of bound states. Following the work of Sokolov we have sketched how a three-particle mass operator with pairwise interactions and correct cluster properties can be constructed. This is accomplished by a set of unitary transformations called packing operators. For the simplest case of three spinless particles we have explicitly calculated these packing operators. In a next step it is planned to use these results to see whether the problems encountered with electromagnetic bound-state currents can be cured by starting with a mass operator that has the correct cluster properties. References 1. V. Bargmann, Ann. Math. 59,1 (1954) 2. B.D. Keister and W.N. Polyzou, Adv. Nucl. Phys. 20, 225 (1991) 3. P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949) 4. B. Bakamjian and L. H. Thomas, Phys. Rev. 92,1300 (1953) 5. S.N. Sokolov and A.N. Shatnii, Theor. Math. Phys. 37,1029 (1978) 6. F. Coester, Helv. Phys. Acta 38, 7 (1965) 7. U. Mutze, J. Math. Phys. 19, 231 (1978) 8. E.P. Biernat, W. Schweiger, K. Fuchsberger and W.H. Klink, Phys. Rev. C 79, 055203 (2009) 9. M. Gomez-Rocha and W. Schweiger, Phys. Rev. D 86, 053010 (2012) 10. E.P. Biernat and W. Schweiger, Phys. Rev. C 89, 055205 (2014) 11. B.D. Keister and W.N. Polyzou, Phys. Rev. C 86, 014002 (2012) 12. S.N. Sokolov, Theor. Math. Phys. 36, 682 (1978) 13. F. Coester and W.N. Polyzou, Phys. Rev. D 26,1348 (1982) Bled Workshops in Physics Vol. 18, No. 1 p. 30 Bled, Slovenia, July 2 - 9, 2017 Baryon Masses and Structures Beyond Valence-Quark Configurations* R.A. Schmidt, W. Plessas, and W. Schweiger Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria Abstract. In order to describe baryon resonances realistically it has turned out that three-quark configurations are not sufficient. Rather explicit couplings to decay channels are needed. This means that additional degrees of freedom must be foreseen. We report results from a study of the nucleon ground state and the Delta resonance by including explicit pionic effects. All current approaches to quantum chromodynamics (QCD) struggle with a proper description of hadron resonances. For baryons one has found that in case of ground states at low energies three-quark configurations can still provide a reasonable picture. For instance, in a relativistic constituent-quark model relying on {QQQ} configurations only, the masses of all ground-state baryons as well as their electromagnetic and axial structures can be well reproduced [1]. In this framework, however, the resonant states are afflicted with severe shortcomings. While the characteristics of the mass spectra can still be yielded to some extent, the reaction properties of baryon resonances fall short, especially with respect to their strong decays. Obviously the reason is that with three-quark configurations only the resonances are described as excited bound states with real eigenvalues rather than genuine resonant states with complex eigenvalues. Consequently, the corresponding wave functions or amplitudes show a completely distinct behaviour. We have started to include beyond {QQQ} configurations explicit mesonic degrees of freedom. In the first instance, we have studied pionic effects in the N and the A masses. We have done so by considering n-loop effects on the hadronic as well as the microscopic quark levels. Our program aims at developing a coupled-channels relativistic constituent-quark model that can generate consistently the strong vertex form factors, the baryon ground-state and resonant masses as well as their electroweak structures. It will contain mesonic degrees of freedom such as {QQQn}, {QQQp}, and eventually {QQQnn} etc. Here we discuss results obtained from n-dressing of the N and the A on the hadronic level. We have investigated the most important one-n-loop effects and several higher-order diagrams. A first account of this study was given already in Ref. [2], where also the formalism and details of the calculation are explained. In this context one has in the first instance to solve an eigenvalue equation, which * Talk presented by W. Plessas results from coupling of a bare N and a bare A to a single n according to the diagrams in Fig. 1. It yields the bare and dressed masses, where the latter is real for the N ground state and becomes complex for the A resonance. The only input into the calculation are the prescriptions for the nNN and nNA form factors at the strong-interaction vertices. For that we have employed models existing in the literature [3-5]. Beyond the results already produced in Ref. [2] we give here in addition values for the dressing effects by using the more recent form-factor parametrization by Kamano et al. [6] derived from a coupled-channels meson-nucleon model. The different form factors are parametrized through the formulae fnni ^ = 1 +( kn y + (kg )4 or F„Ni = exp-^2^ or FnNI ^r+72) > (1) where B stands either for N or A. The values of the various cut-off parameters are given in Tab. 1 together with the corresponding coupling constants. 2 2 Table 1. Parameters of the bare nNIN and nN A vertex form factors. The first three columns correspond to the multipole type as in the first formula of Eq. (1), the fourth column to the Gaussian type as in the second formula of Eq. (1), and the last column to the dipole type as in the third formula of Eq. (1). The corresponding parametrizations are taken from Refs. [3], [5] and [6], respectively. All (bare) coupling constants belong to k^ = 0. RCQM refers to the predictions of the relativistic constituent-quark model [7] in Ref. [3], SL to the nN meson-exchange model by Sato and Lee [4], PR to the Nijmegen soft-core model of Polinder and Rijken [5], and KNLS to the coupled-channels meson-nucleon model of Kamano, Nakamura, Lee, and Sato. All cut-off parameters are in GeV. RCQM SL PR multipole PR Gaussian KNLS f2 /4n 0.0691 nN N' 0.08 0.013 0.013 0.08 AT 0.451 nNN A2 0.931 A 0.453 0.940 0.641 1.102 0.665 0.656 f2 ^/4n 0.188 nN A' 0.334 0.167 0.167 0.126 At 0.594 nNÀ A2 0.998 A 0.458 0.853 0.648 1.014 0.603 0.709 For the nN N vertex the momentum dependences of the form factors from the five different models are shown in Fig. 2. With these ingredients the n-dressing effects in the N mass are yielded as in Tab. 2. The mass shifts are basically of the same order of magnitude for all form-factor models employed, even though n n Fig. 1. n-loop diagrams considered for the dressing of a bare N and a bare A. nNN Form Factor Fig. 2. Dependences on the n three-momentum squared k^ of the different (bare) form-factor models for the nNNN system. the momentum dependences are quite different as seen from Fig. 2. However, the net effect is gained from an interplay of the momentum dependence of each form factor and the corresponding nNNN coupling constant (cf. Tab. 1). The largest dressing effect is obtained in case of the RCQM. Table 2. n-loop effects in the N mass mN = 0.939 GeV according to the l.h.s. diagram of Fig. 1. RCQM SL PR multipole PR Gaussian KNLS mN [GeV] 1.067 1.031 1.051 1.025 1.037 mN - mN [GeV] 0.128 0.092 0.112 0.086 0.098 For the nNN A vertex the momentum dependences of the form factors from the five different models are shown in Fig. 3. With these ingredients the n-dressing effects in the A mass are yielded as in Tab. 3. It is immediately evident that the A mass gets complex. The real part corresponds to resonance position in the nN channel and the complex part to (half) the hadronic A decay width. While the n-dressing effects in the real part are of about the same magnitudes as in the N, in all cases the decay widths are much too small as compared to the empirical value of about 0.117 GeV. 1.0 0.8 rCT 0.6 t ll_ 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 kn 2[GeV2] Fig. 3. Dependences on the n three-momentum squared k^ of the different (bare) form-factor models for the nN A system. Table 3. n-loop effects in the A mass Re (m.A )= 1.232 GeV and in the n-decay width r according to the r.h.s. diagram of Fig. 1, where the bare N masses m^ in the intermediate states are the same as in Table 2. RCQM SL PR multipole PR Gaussian KNLS mA [GeV] 1.300 1.290 1.335 1.321 1.259 mA - Re (mA) [GeV] 0.068 0.058 0.103 0.089 0.027 r = 2Im (mA) [GeV] 0.004 0.023 0.008 0.016 0.007 An improvement in the A —» nN decay width F is achieved by replacing the bare N in the intermediate state by the dressed N like in Fig. 4. Thereby the phase space for the strong decay is enlarged, and the situation may be closer to the realistic one. The n-dressing effect in the real part is slightly raised in all cases, as compared to the values in Tab. 3, however, the changes achieved for the decay width F are respectable. Now, they reach about 50% of the phenomenolog-ical value, except for the KNLS form-factor model. Still, the results appear to be unsatisfactory. Therefore we have investigated higher-order effects, i.e. two-n loops, where the ones with n-n interactions in the intermediate state can be effectively described by d and p mesons. The corresponding dressing effects turned to be marginal. Their inclusions do not help much to improve the A decay width. nNA Form Factor 1 V — — RCQM » X SL \\ PR Multipole PR Gauss \ \ \x ........ KNLS \ \ *V v. >s N ^ • — - - n ~ N ~ Fig. 4. n-loop diagram considered for the dressing of a bare A, where in the intermediate state a physical N with mass mN =0.939 GeV is employed. Table 4. n-loop effects in the A mass Re (mi)= 1.232 GeV and in the n-decay width r according to the diagram in Fig. 4, where in the intermediate state always mN = 0.939 GeV. RCQM SL PR multipole PR Gaussian KNLS mA [GeV] 1.309 1.288 1.347 1.328 1261 m^ - Re (mA) [GeV] 0.077 0.056 0.114 0.096 0.029 r = 2Im (mA) [GeV] 0.047 0.064 0.052 0.051 0.027 We are now in the course of investigating explicit pionic effects on the microscopic level, i.e. along a relativistic coupled-channels constituent-quark model. This will also help us to get rid of inputs of vertex form factors foreign to the quark model, because in such an approach one can determine within the same framework both the mass dressings as well as the vertex form factors consistently. Acknowledgment The authors are grateful to Bojan Golli, Mitja Rosina, and Simon Sirca for their continuous efforts of organizing every year the Bled Mini-Workshops. These meetings serve as a valuable institution of exchanging ideas and of mutual learning among an ever growing community of participating colleagues engaged in hadronic physics. This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16). References 1. W. Plessas, Int. J. Mod. Phys. A 30,1530013 (2015) 2. R. A. Schmidt, L. Canton, W. Plessas, and W. Schweiger, Few-Body Syst. 58, 34 (2017) 3. T. Melde, L. Canton, and W. Plessas, Phys. Rev. Lett. 102,132002 (2009) 4. T. Sato and T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996) 5. H. Polinder and T. A. Rijken, Phys. Rev. C 72, 065210 (2005); ibid. 065211 6. H. Kamano, S. X. Nakamura, T.-S. H. Lee, and T. Sato, Phys. Rev. C 88, 035209 (2013) 7. L. Y. Glozman, W. Plessas, K. Varga and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998) Bled Workshops in Physics Vol. 18, No. 1 p. 35 Bled, Slovenia, July 2 - 9, 2017 Single energy partial wave analyses on eta photoproduction - experimental data J. Stahov*a, H. Osmanovica, M. Hadzimehmedovica, R. Omerovica, V. Kashevarovb, K. Nikonovb, M. Ostrickb, L. Tiatorb, A. Svarcc aUniversity of Tuzla, Faculty of Natural Sciences and Mathematics, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina bInstitut fur Kernphysik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany cRudjer Boskovic Institute, Bijenicka cesta 54, P.O. Box 180,10002 Zagreb, Croatia Abstract. Free, unconstrained, single channel, single energy partial wave analysis of r| photoproduction is discontinuous in energy. We achieve point-to-point continuity by enforcing fixed-t analyticity on model independent way using available experimental data, and show that present database is insufficient to produce a unique solution. The fixed-t analyticity in the fixed-t amplitude analysis is imposed by using Pietarinen's expansion method known from Karlsruhe-Helsinki analysis of pion-nucleon scattering data. We present an analytically constrained partial wave analysis using experimental data for four observables recently measured at MAMI and GRAAL in the energy range from threshold to /s = 1.85 GeV. 1 Introduction In another contribution of our group [1] to the Mini Workshop, we applied iterative procedure with the fixed-t analyticity constraints to a partial wave analysis of eta photoproduction pseudo data. In this paper we apply our method to a partial wave analysis of experimental data considering some limitations due to use of real data instead of idealised pseudo one. Presently, we have an incomplete set of experimental data consisting of differential cross section 00, single target polarisation asymmetry T, double beam-target polarisation with circular polarized photons F, and single beam polarisation I. Statistical and systematic errors of experimental data are much larger than 0.1% used in our analysis with pseudodata. There is also limitation in kinematical coverage. Unpolarized differential cross section has the best coverage in energy and scattering angles. Good coverage is also available for the polarisation data (I, T, F) up to total c.m. energy W = 1.85GeV. More details about formalism and our method may be found in ref. [2]. * Talk presented by J. Stahov 36 J. Stahov et al. 2 Input preparation and results The list of data which we used in our PWA analysis with experimental data is given in Table 1. Table 1. Experimental data from A2@MAMI and GRAAL used in our PWA. Obs N Eiab [MeV] Ne Scm [0] Ne Reference oo 2400 710 — 1395 120 18 — 162 20 A2@MAMI(2010,2017) [3,4] I 150 724 — 1472 15 40 — 160 10 GRAAL(2007) [5] T 144 725 — 1350 12 24 — 156 12 A2@MAMI(2014) [6] F 144 725 — 1350 12 24 — 156 12 A2@MAMI(2014) [6] As it may be seen from the table, in our data base we have data for differential cross sections at much more energies then for polarisation observables. To perform partial wave analysis, all observables are needed at the same kine-matical points. Experimental values of double-polarisation asymmetry F, target asymmetry T, and beam asymmetry I for given scattering angles have to be interpolated to the energies where the ct0 data are available ( fixed-s data binning). We have used a spline smoothing method as a standard method for interpolation and data smoothing [7] (Fortran code available on request). In the Ft AA part of our method, we have to build a data base at fixed t-values using measured angular distribution at a fixed value of variable s (fixed-t data binning). This has been done using again spline interpolation and smoothing method. We have performed Ft AA at t-values in the range —1.00GeV2 < t < —0.09GeV2 at 20 equidistant values. When working with real data, uniqueness means that the partial wave solution does not depend on starting solution. We start with two different MAID solutions: Solution I (EtaMAID-2016, [8]) and Solution II (EtaMAID-2017, [3]). Although significantly different, both solutions describe experimental data very well as might be seen in Figure 1 for two values of variable t (predictions from these two solutions can not be distinguished in the plots). In our truncated PWA we fitted partial waves up to Lmax=5. As in the case of pseudo data, procedure has converged fast. Resulting multipoles up to L=2, obtained after three iterations, are shown in Figure 2. Almost no differences can be observed for the dominant S waves, what is to be expected while this wave is similar in both starting solutions. Other partial waves are consistent within their error bands. Considerable differences still exist in certain kinematical regions, mainly at higher energies. It is a strong indication that for some multipoles a unique solution in this kinematical regions was not achieved (See ImEi+, ImE2-, and ReM-2- for example). There are different reasons for nonuniqueness observed. First of all, we have as an input an incomplete set of four observables. Secondly, our fixed- t constraint loses its constraining power at higher energies, especially at larger scattering angles. In addition, less kinematical points are experimentally accessible for higher negative t- values. From partial wave analysis Single energy partial wave analyses on eta photoproduction- experimental data 37 1.5 G 1 o 0.5 0 0.2 G "O o 0.1 "O I— 0 Fig. 1. Pietarinen fit of the interpolated data at t = -0.2GeV2 and t = -0.5 GeV2. The dashed (black) and solid (blue) curves are the results starting with solutions I and II respectively and are on top of each other. 1500 1650 1800 W[MeV] 1500 1650 1800 W[MeV] 1500 1650 1800 W[MeV] 1500 1650 1800 W[MeV] E-4 -12 1 0.6 LLI10 1450 1650 1850 1450 1650 1850 W [MeV] W [MeV] w 0 e tr -0.6 1450 1650 1850 -5 1 0.6 E 0.15 E -0.3 1.6 - 0 1 2.5 i fI|i:t 1 -0.2 1450 1650 1850 1450 1650 1850 W [MeV] W [MeV] «H ^ 1.5 - ¡10.25 -1.5 ♦v !t W [MeV] 1450 1650 1850 -0 5450 1650 1850 1450 1650 1850 W [MeV] W [MeV] W [MeV] tK 0 -0.15 0.2 1 0.2 Si ~ ^ LLJ 0.05 i-0.4 -1 _ -0.1 1450 1650 1850 1450 1650 1850 W [MeV] 1 2 W [MeV] , 0.15 fSs . 0.4 ¥ -0.4 -0.2 0.3 5-0.05 _ -0.4 1450 1650 1850 1450 1650 1850 W [MeV] 06 W [MeV] i E -0.2 - ••• *ïi M 0.1 ^ £ E . ... - -0.8-1- -0.4 -1- 1450 1650 1850 1450 1650 1850 1450 1650 1850 1450 1650 1850 W [MeV] W [MeV] W [MeV] W [MeV] 4 25 Fig. 2. (Color online) Real and imaginary parts of the S-, P- and D-wave multipoles obtained in the final step after three iterations using analytical constraints from helicity amplitudes obtained from initial solutions I (blue) and II (red). of pseudo data we learned that a complete set of data results in unique solution. From that reason, we presume that new data from ELSA, JLAB and MAMI, which are expected soon, will help to resolve remaining ambiguities. 3 Conclusions We applied iterative procedure with the fixed-t analyticity constraints to a partial wave analysis of eta photoproduction experimental data. In truncated PWA we obtained multipoles up to Lmax=5. Ambiguities still remain in some multipoles, mainly at higher energies. New data, expected soon, will significantly expand our database, improve reliability of our results, and resolve remaining ambiguities. References 1. H. Osmanovic, M. Hadzimehmedovic, R. Omerovic, S. Smajic, J. Stahov,V. Kashe-varov, K. Nikonov, M. Ostrick, L. Tiator, and A. Svarc, paralel publication in the same issue (Proceedings-Bled2017). 2. H. Osmanovic, M. Hadzimehmedovic, R. Omerovic, J. Stahov,V. Kashevarov, K. Nikonov, M. Ostrick, L. Tiator, and A. Svarc, arXiv:1707.07891 [nucl-th] 3. V. L. Kashevarov et al., Phys. Rev. Lett. 118, 212001 (2017). 4. E. F. McNicoll et al. [Crystal Ball at MAMI Collaboration], Phys. Rev. C 82, 035208 (2010). Erratum: [Phys. Rev. C 84, 029901 (2011)] 5. O. Bartalini et al. [GRAAL Collaboration], Eur. Phys. J. A 33,169 (2007). 6. C. S. Akondi et al, [A2 Collaboration at MAMI], Phys. Rev. Lett. 113,102001 (2014). 7. C. de Boor, A Practical Guide to Splines, Springer-Verlag, Heidelberg, 1978, revised 2001. 8. V. L. Kashevarov, L. Tiator, M. Ostrick, JPS Conf. Proc. 13, 020029 (2017). Bled Workshops in Physics Vol. 18, No. 1 p. 39 Bled, Slovenia, July 2 - 9, 2017 Exclusive pion photoproduction on bound neutrons I. Strakovsky The George Washington University Abstract. An overview of the GW SAID group effort to analyze pion photoproduction on the neutron target was given. The disentangling of the isoscalar and isovector EM couplings of N* and A* resonances requires compatible data on both proton and neutron targets. The final-state interactions play a critical role in the state-of-the-art analysis in extraction of the yu —> nN data from the deuteron target experiments. Then resonance couplings determined by the SAID PWA technique are compared to previous findings. The neutron program is an important component of the current JLab, MAMI-C, SPring-8, ELSA, and ELPH studies. This research is supported in part by the US Department of Energy under Grant No. DE-SC0016583. Bled Workshops in Physics Vol. 18, No. 1 p. 40 Bled, Slovenia, July 2 - 9, 2017 Resonances and strength functions of few-body systems Y. Suzuki Department of Physics, Niigata University, Niigata 950-2181, Japan RIKEN Nishina Center, Wako 351-0198, Japan Abstract. A resonance offers a testing ground for few-body dynamics. Two types of resonances are discussed in detail. One is very narrow Hoyle resonance in 12 C that plays a crucial role in producing that element in stars. The other includes broad high-lying negative-parity resonances in A = 4 nuclei, 4H, 4He, 4Li. The former is dominated by the Coulomb force of three-a particles at large distances, while the latter are by short-ranged nuclear forces. The structure of these resonances is described by different approaches, adiabatic hyperspherical method and correlated Gaussians used for strength function calculations. The localization of the resonance is successfully realized by a complex absorbing potential and a complex scaling method, respectively. 1 Hoyle resonance The synthesis of 12C is essential to 12C-based life and its process at low temperatures is sequential via a narrow resonance of 8Be: As predicted by Hoyle, however, an existence of a very narrow resonance at around Ex =7.7 MeV is vital to explain the abundance of 12C element. The resonance is found to be just 0.38 MeV above 3a threshold with its width of 8.5 eV. Since nobody has ever succeeded in reproducing the Hoyle resonance width, we have undertaken to tackle this problem in the adiabatic hyperspherical method [1, 2]. This study has further been motivated by the fact that there exists huge discrepancy in the rate of triple-a reactions, a + a + a —» 12C + y, calculated by several authors [3-5]. In contrast to two-body resonances, the Hoyle resonance is characterized by the followings: (1) 3a particles interact via long-ranged Coulomb force even at large distances. (2) no asymptotic wave function is known. (3) 2a subsystem forms a sharp resonance, which causes successive avoided crossings with three-particle continuum states. The detail of our approach is given in Refs. [1,2]. The three-body system is completely specified by six coordinates excluding the center-of-mass coordinate. Among six coordinates one is chosen to be the hyperradius of length dimension, and other five coordinates are hyperangles. Among the five angle coordinates a + a^ 8Be, a + 8Be ^ 12C + y. (1) three are Euler angles and two are used to specify the geometry of the three body system. By changing the geometry as much as possible, we can study the adia-batic potential curve of the three-body system as a function of the hyperradius. A resonance can be confined by introducing a complex absorbing potential [6] at large distances of the hyperradius. This method works excellently for quantitatively reproducing the very narrow width of the Hoyle resonance as well as predicting the triple-a reaction rate at low temperatures without relying on any ambiguous ansatz. 2 Resonances in A = 4 nuclei 4He nucleus is doubly magic and its 0+ ground state is strongly bound. The first excited state of 4He is not a negative-party but again 0+. The negative-parity excited states appear above the 3He+p threshold. Seven negative-parity states are known and some of them have very broad widths. There exist isobar resonances in 4H and 4Li that are also very broad. Most of these resonances are identified by R-matrix phenomenology. These resonances offer typical four-body resonances governed by the nuclear force. The decay channels include not only two-body but three-body systems. To describe the resonance we have employed correlated Gaussians [7,8] that provide us with efficient and accurate performance as few-body basis functions. A general form of the correlated Gaussians is [0l x xs]jm exp —Y_ aij(ri — rj)2 nTMy, (2) i A(W, 9) = e 1*(W'e) CO x + 1)A«(W)P«(cos 9) «=0 CO A (W, 9) = + 1)A «(W )P«(cos 9) (3) «=0 We are interested in rotated partial wave amplitudes A«(W), defined by Eq.(3), and are free to introduce the Legendre decomposition of an exponential function as: CO e 1*(W>e) = ^ L«(W)P«(cos9). «=0 After some manipulation of the product Pt(x)Pk(x) (see refs. [6,7] for details of the summation rearrangement) we obtain: CO t ' + t At(W) = ^ Lt'(W) • ^ (I',0;£,0|m,0)2 Am(W) t '=0 m=|t '-t| (5) where (£', 0; £, 0|m, 0) is a standard Clebsch-Gordan coefficient. To get a better insight into the mechanism of multipole mixing, let us expand Eq. (5) in terms of phase-rotation Legendre coefficients Lt ' (W): A o(W) A i(W) A 2(W) The consequence of Eqs. (5) and (6) is that angular-dependent phase rotations mix multipoles. Conclusion: Without fixing the free continuum ambiguity phase ^ (W, 9), the partial wave decomposition A^ (W) defined in Eq. (2) is non-unique. Partial waves get mixed, and identification of resonance quantum numbers might be changed. To compare different partial-wave analyses, it is essential to match the continuum ambiguity phase; otherwise the mixing of multipoles is yet another, uncontrolled, source of systematic errors. Observe that this phase rotation does not create new pole positions, but just reshuffles the existing ones among several partial waves. 3 Using angular-dependent phase ambiguity to obtain up-to-a-phase unique, unconstrained, single-energy solution in n photoproduction We perform unconstrained, Lmax = 5 truncated single-energy analyses on a complete set of observables for n photoproduction given in the form of pseudodata created using the ETA-MAID15a model [8]: do/dn, I do/dn, T do/dn, F do/dn, G do/dn, P do/dn, Cx/ do/dn, and Ox/ do/dn. All higher multipoles L0(W)A0(W) + Lt (W)At (W) + L2(W)A2(W) + ..., L>(W)Ai(W) + Lt (W) (6) 3Ao(W ) + 2A2(W) +L2(W ) 2 3 5At(W) + 3Aa(W) +..., Lo(W)A2(W) + Lt(W) + L2(W ) 23 5At(W ) + 3Aa(W) 1 2 18 -Ao(W) + 2A2(W) + i8A4(W) 5 7 35 + .... are put to zero. The fitting procedure finds solutions which are non-unique, and we obtain many solutions depending on the choice of initial parameters in the fit. In Fig. 1 we show a complete set of pseudo-data with the error of 1 % created at 18 angles (red symbols), and the typical SE fit (full line) at one representative energy of W = 1769.80 MeV. Fig. 1. (Color online) Complete set of observables for r| photoproduction given in the form of pseudo-data created at 18 angles with the error bar of 1 % using the ETA-MAID15a model (red symbols) and a typical fit to the data (full line). In Fig. 2 we show an example of three very different sets of multipoles which fit the complete pseudo-data set equally well to a high precision: two discrete and discontinuous ones obtained by setting the initial fitting values to the ETA-MAID16a [9] (SE16a) and Bonn-Gatchina [10] (SEBG) model values (blue and red symbols respectively), and the generating ETA-MAID15a model [8] which is displayed as full and dashed black continuous lines. We know from Eq.(1) that equivalent fits to a complete set of data must be produced by helicity amplitudes with different phases. Therefore, in Fig. 3, we construct the helicity amplitudes corresponding to all three sets of multipoles from Fig. 2 at one randomly chosen energy W = 1660.4 MeV. We see that all three sets of helicity amplitudes are indeed different, but the discontinuity of multipole amplitudes, observed in Fig. 2-left is not reflected in a plot of helicity amplitudes at a fixed single energy. If instead we plot an excitation curve of all four helicity amplitudes at a randomly chosen angle, which is arbitrarily set to the value cos 0 = 0.2588, we obtain the result shown in Fig. 3-right. We see that the excitation curve of helicity amplitudes in this case remains continuous only for the generating model ETA-MAID15a. For both single-energy solutions it is different, and at the same time shows notable discontinuities between neighbouring energy points. This leads to the following understanding of this, apparently very different multipole solutions: W[MeV] W[MeV] Fig. 2. (Color online) Plots of the Eo+, Mi_, Ei +, and Mi + multipoles. Full and dashed black lines give the real and imaginary part of the ETA-MAID15a generating model. Discrete blue and red symbols are obtained with the unconstrained, Lmax = 5 fits of a complete set of observables generated as numeric data from the ETA-MAID15a model of ref. [8], with the initial fitting values taken from the ETA-MAID16a [9] and the Bonn-Gatchina [10] models respectively. Filled symbols represent the real parts and open symbols give the imaginary parts. E, 0 nf -2 _ 5 E 1 0 CO X -5 •"o « * o \ • 0 •»ÎÎÎ J O 0 0 oO° ° 0 ° ° o e O 8-5S- » s : V Vo 8 . • 0 cos e 2100 1500 1800 2100 20 E 10 E n _L -10 \ .A Uv ___ w Oo o o 1500 1800 2100 W [MeV] 1500 1800 2100 W [MeV] Fig. 3. (Color online) Left we show three sets of helicity amplitudes for all three sets of multipoles at one randomly chosen energy W = 1660.4 MeV, and right for we show the excitation curves for all three sets of multipoles, at one randomly chosen value of cos 6 = 0.2588 MeV. The figure coding is the same as in Fig. 2. When we perform an unconstrained SE PWA, each minimization is performed independently at individual energies, and the phase is chosen randomly. So, at each energy the fit chooses a different angle dependent phase, and creates different, discontinuous numerical values for each helicity amplitude, producing discontinuous sets of multipoles. However, the invariance with respect to phase rotations offers a possible solution. Let us show the procedure. We introduce the following angle-dependent phase rotation simultaneously for all four helicity amplitudes: HkE(W,0) = HSE(W,0) ■ eiOH2a(W'0)-^(W'e) k = 1,... ,4 (7) where (W, 0) is the phase of any single-energy solution and ®|f2a(W, 0) is the phase of generating solution ETA-MAID15a. Applying this rotation we replace the discontinuous Off (W 0) phase from any SE solution with the continuous ®ff2a(W, 0) ETA-MAID15a phase. 1500 1800 2100 1500 1800 2100 1500 1800 2100 W[MeV] 1500 1800 2100 W[MeV] Fig. 4. (Color online) Up we show all three sets of rotated helicity amplitudes at one randomly chosen energy W = 1660.4 MeV, and down three sets of rotated multipoles. The figure coding is the same as in Fig. 2. The resulting rotated single-energy helicity amplitudes are compared with generating ETA-MAID15a amplitudes in Fig. 4. We see that rotated helicity amplitudes of both single-energy solutions are now identical to the generating ETA-MAID15a helicity amplitudes. Thus, the previously different sets of discrete, discontinuous single-energy multipoles different from the generating solution ETA-MAID15a and given in Fig. 2, are after phase rotation transformed into continuous multipoles now identical to the generating solution, and given in lower part of Fig. 4. So, we have constructed a way to generate up-to-a-phase unique solutions in an unconstrained PWA of a complete set of observables generated as pseudodata. 4 Laurent + Pietarinen expansion The driving concept behind the Laurent-Pietarinen (L+P) expansion was the aim to replace an elaborate theoretical model by a local power-series representation of partial wave amplitudes [2]. The complexity of a partial-wave analysis model is thus replaced by much simpler model-independent expansion which just exploits analyticity and unitarity. The L+P approach separates pole and regular part in the form of a Laurent expansion, and instead of modeling the regular part in some physical model it uses the conformal mapping to expand it into a rapidly converging power series with well defined analytic properties. So, the method replaces the regular part calculated in a model by the simplest analytic function which has correct analytic properties of the analyzed partial wave (multipole), and fits the data. In such an approach the model dependence is minimized, and is reduced to the choice of the number and location of branch-points used in the model. The L+P expansion is based on the Pietarinen expansion used in some former papers in the analysis of pion-nucleon scattering data [11-14], but for the L+P model the Pietarinen expansion is applied in a different manner. It exploits the Mittag-Leffler expansion1 of partial wave amplitudes near the real energy axis, representing the regular, but unknown, background term by a conformal-mapping-generated, rapidly converging power series called a Pietarinen expan-sion2. The method was used successfully in several few-body reactions [3,4,17], and recently generalized to the multi-channel case [18]. The formulae used in the L+P approach are collected in Table 1. In the fits, the regular background part is represented by three Pietarinen expansion series, all free parameters are fitted. The first Pietarinen expansion with branch-point xP is restricted to an un-physical energy range and represents all left-hand cut contributions. The next two Pietarinen expansions describe the background in the physical range with branch-points xq and xR respecting the analytic properties of the analyzed partial wave. The second branch-point is mostly fixed to the elastic channel branchpoint, the third one is either fixed to the dominant channel threshold, or left free. Thus, only rather general physical assumptions about the analytic properties are made like the number of poles and the number and the position of branch-points, and the simplest analytic function with a set of poles and branch-points is constructed. The method is applicable to both, theoretical and experimental input, and represents the first reliable procedure to extract pole positions from experimental data, with minimal model bias. The generalization of the L+P method to a multichannel L+P method is performed in the following way: i) separate Laurent expansions are made for each channel; ii) pole positions are fixed for all channels, iii) residua and Pietarinen coefficients are varied freely; iv) branch-points are chosen as for the single-channel model; v) the single-channel discrepancy function D^p (see Eq. (5) in ref. [17]) which quantifies the deviation of the fitted function from the input is generalized to a multi-channel quantity D by summing up all single-channel contributions, and vi) the minimization is performed for all channels in order to obtain the final solution. The formulae used in the L+P approach are collected in Table 1. 1 Mittag-Leffler expansion [15]. This expansion is the generalization of a Laurent expansion to a more-than-one pole situation. For simplicity, we will simply refer to this as a Laurent expansion. 2 A conformal mapping expansion of this particular type was introduced by Ciulli and Fisher [11,12], was described in detail and used in pion-nucleon scattering by Esco Pietarinen [13,14]. The procedure was denoted as a Pietarinen expansion by G. Hohler in [16]. Table 1. Formulae defining the Laurent+Pietarinen (L+P) expansion. N pole a + a Ka La Ma Ta(W)= X XW +-W + X Ca Xa(W)k + X da Ya(W)' + X Za(W)m j=1 j k=0 1=0 m=0 a, , ac ^/xS - W a, ^ |3a - VxQ - W Ya ^/xS - W Xa(W) = -v P ; Ya(W) _ --v, Q ; Za(W) _ --v R ac Wxg - W Pa WxQ - W ' ya WxRa - W Da _ 1 V" Ddp 2 N a - Na NW ReTa(W(i)) - ReTa'exp(W(i)) + ImTa(W(i)) -ImTa>exp(W(i)) Errím i,a .Re 2 " ErrR i,a + P 0 Ka La Ma all Pa _ Aca ^(cs)2 k3 + Ad X(d°a)2 l3 + AC X (em)2 m3 ; Ddp _ X DCp k=1 l=1 m=1 a a ... channel index Npole ... number of poles Wj,W e C a a a i a a a o a a ,-rni xa, Wa, Ca, da, em, a , p , Y ... e R Ka, La, Ma ... e N number of Pietarinen coefficients in channel a. D o ... discrepancy function in channel a NW ... number of energies in channel a NCar ... number of fitting parameters in channel a Pa . . . Pietarinen penalty function Ac, Ac, Ac ... Pietarinen weighting factors xC, xQ, xC e R (or e C). ErrR,CIm... minimization error of real and imaginary part respectively. Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft SFB 1044. References 1. A. Svarc, https://indico.cern.ch/event/591374/contributions/2477135/ , PWA9/ATHOS4: The International Workshop on Partial Wave Analyses and Advanced Tools for Hadron Spectroscopy, Bad Honnef near Bonn (Germany) from March 13 to 17, 2017. 2. A. Svarc, M. Hadzimehmedovic, H. Osmanovic, J. Stahov, L. Tiator, and R. L. Workman, Phys, Rev. C88, 035206 (2013). 3. A. Svarc, M. Hadzimehmedovic, R. Omerovic, H. Osmanovic, and J. Stahov, Phys, Rev. C89, 0452205 (2014). 4. A. Svarc, M. Hadzimehmedovic, H. Osmanovic, J. Stahov, L. Tiator, and R. L. Workman, Phys, Rev. C89, 65208 (2014). 5. L. Tiator, D. Drechsel, S. S. Kamalov and M. Vanderhaeghen, Eur. Phys. J. ST 198, 141 (2011). 6. J. Dougall, Glasgow Mathematical Journal, 1 (1952) 121-125. 2 7. Y. Wunderlich, A. Svarc, R. L. Workman, L. Tiator, and R. Beck, arXiv:1708.06840[nucl-th]. 8. V. L. Kashevarov, L. Tiator, M. Ostrick, Bled Workshops Phys., 16, 9 (2015). 9. V. L. Kashevarov, l. Tiator, M. Ostrick, JPS Conf. Proc. 13, 020029 (2017). 10. http://pwa.hiskp.uni-bonn.de/ 11. S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961). 12. I. Ciulli, S. Ciulli, and J. Fisher, Nuovo Cimento 23,1129 (1962). 13. E. Pietarinen, Nuovo Cimento Soc. Ital. Fis. 12A, 522 (1972). 14. E. Pietarinen, Nucl. Phys. B107, 21 (1976). 15. Michiel Hazewinkel: Encyclopaedia of Mathematics, Vol.6, Springer, 31. 8.1990, pg.251. 16. G. Hohler and H. Schopper, "Numerical Data And Functional Relationships In Science And Technology. Group I: Nuclear And Particle Physics. Vol. 9: Elastic And Charge Exchange Scattering Of Elementary Particles. B: Pion Nucleon Scattering. Pt. 2: Methods And Results and Phenomenology," Berlin, Germany: Springer ( 1983) 601 P. ( Landolt-Boernstein. New Series, I/9B2). 17. A. Svarc, M. HadZimehmedovic, H. Osmanovic, J. Stahov, and R. L. Workman, Phys. Rev. C91, 015207 (2015). 18. A. Svarc, M. HadZimehmedovic, H. Osmanovic, J. Stahov, L. Tiator, R. L. Workman, Phys. Lett. B755 (2016) 452455. Bled Workshops in Physics Vol. 18, No. 1 p. 52 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 Baryon transition form factors from space-like into time-like regions L. Tiator Institut für Kernphysik, Johannes Gutenberg Universität Mainz, Germany Pion electroproduction is the main source for investigations of the transition form factors of the nucleon to excited N* and A baryons. After early measurements of the GM form factor of the NA transition, in the 1990s a large program was running at Mainz, Bonn, Bates Brookhaven and JLab in order to measure the E/M ratio of the NA transition and the Q2 dependence of the E/M and S/M ratios in order to get information on the internal quadrupole deformations of the nucleon and the A. Only at JLab both the energy and the photon virtuality were available to measure transition form factors for a set of nucleon resonances up to Q2 « 5 GeV2. Two review articles on the electromagnetic excitation of nucleon resonances, which give a very good overview over experiment and theory, were published a few years ago [1,2]. In the spirit of the dynamical approach to pion photo- and electroproduction, the t-matrix of the unitary isobar model MAID is set up by the ansatz [1] tyn(W) = t£„(W)+ tR„(W) , (1) with a background and a resonance t-matrix, each of them constructed in a unitary way. Of course, this ansatz is not unique. However, it is a very important prerequisite to clearly separate resonance and background amplitudes within a Breit-Wigner concept also for higher and overlapping resonances. For a specific partial wave a = {j, l,...}, the background t-matrix is set up by a potential multiplied by the pion-nucleon scattering amplitude in accordance with the K-matrix approximation, tBna(W, q2) = vBna(W, Q2) [1 + Ne+e-. At nN threshold, the pseudo-threshold value is Qpt = -mn = -0.018 GeV2, at W = 1535 MeV, Q^t = -0.356 GeV2. The vertical lines denote the pion threshold and nucleon resonance positions, where space-like transition form factors have been analyzed from, electroproduction experiments. For the resonance contributions we follow Ref. [3] and assume Breit-Wigner forms for the resonance shape, tr,afwq21= tr (wq2n fyn(w)rtot(W) mr fnn (W) ^ (w) (3) (w,q Aa (w,q n mr - w2 - imr rtot(w) e , (3) where fnN (W) is a Breit-Wigner factor describing the decay of a resonance with total width rtot (W). The energy dependence of the partial widths and of the yNN* vertex can be found in Ref. [3]. The phase (W) in Eq. (3) is introduced to adjust the total phase such that the Fermi-Watson theorem is fulfilled below two-pion threshold. In most cases, the resonance couplings Aa(W, Q2) are assumed to be independent of the total energy and a simple Q2 dependence is assumed for Aa (Q2). Generally, these resonance couplings, taken at the Breit-Wigner mass W = MR are called transition form factors Aa (Q2). In the literature, baryon transition form factors are defined in three different ways as helicity form factors ai/2 (Q2), A3/2(Q2), S1/2(Q2), Dirac form factors F] (Q2), F2(Q2), F3(Q2) and Sachs form factors G e (Q2), G m(Q2), G c (Q2). For detailed relations among them see Ref. [1]. In MAID they are parameterized in an ansatz with polynomials and exponentials, where the free parameters are determined in a fit to the world data of pion photo-and electroproduction. In the case of the NA transition, the form factors are usually discussed in the Sachs definition and are denoted by G|(Q2),Gm(Q2),Gc(Q2). While the Gm form factor by far dominates the N —> A transition, the electric and Coulomb transitions are usually presented as E/M and S/M ratios. In pion electroproduc-tion they are defined as the ratios of the multipoles. Within our ansatz they can be expressed in terms of the NA transition form factors by rem l.q2) = — hfj , (4) k(MA,Q2) GC(Q2) 2ma g*m(q2) , rsm(q2) = ^';;^ j t^cq^ , (5) with the virtual photon 3-momentum k(W, Q2) _ 0(W_ Mn)2 + Q2)(W + Mn)2 + Q2)/(2W) . Q2 (GeV/c)2 Q2 (GeV/c)2 Fig. 2. The Q2 dependence of the E/M and S/M ratios of the A(1232) excitation for low Q2. The data are from Mainz, Bonn, Bates and JLab. For details see Ref. [1]. The behavior of the S/M ratio at low Q2 and in particular for Q2 <0 in the unphysical region is a consequence of the Siegert theorem. Whereas in photo- and electroproduction, data are only available for spacelike momentum transfers, Q2 = —k^k^ > 0, the inelastic form factors can be extended into the time-like region, Q2 < 0, down to the so-called pseudo-threshold, Qpt, which is defined as the momentum transfer, where the 3-momentum of the virtual photon vanishes, k(W, Qpt) = 0 -> Qpt = —(W — mn)2. (6) This time-like region is called the Dalitz decay region. The energy dependence of this region is shown in Fig. 1. At pion threshold, the Dalitz decay region is very small and extends only down to Qpt = —0.018 GeV2, for transitions to the A(1232) resonance down to —0.086 GeV2 and to the Roper resonance N(1440) down to —0.252 GeV2. In Fig. 2 we have extended our parametrization of the E/M and S/M ratios for N —} A from space-like to time-like regions and show a comparison to the data obtained from photo- and electroproduction [1,2]. In general, the extrapolation of the transverse form factors GE and GM into the time-like region is more reliable than the extrapolation of the longitudinal form factor GC, which can not be measured at Q2 = 0 with photoproduction. For longitudinal transitions, the photon point is only reachable asymptotically, and in practise, only at MAMI-A1 in Mainz, momentum transfers as low as Q2 ~ 0.05 GeV2 are accessible. Therefore, the longitudinal form factors become already quite uncertain in the real-photon limit Q2 = 0. Because of this practical limitation, the Siegert Theorem, already derived in the 1930s, give a powerful constraint for longitudinal form factors. In the long-wavelength limit, where k 0, all three components of the e.m. current become identical, Jx = Jy = Jz, because of rotational symmetry. As a consequence, excitations as N —» A(1232)3/2+ or N —» N(1535) 1/2- obtain charge form factors that are proportional to the electric form factors. For the N —» N (1440) 1/2+ transition, where no electric form factor exists, still a minimal constraint remains, namely Si/2 (Q2) - k(Q2), (7) forcing the longitudinal helicity form factor to vanish at the pseudo-threshold. This is a requirement for all S1/2 transition form factors to any nucleon resonance. In Fig. 3 the longitudinal transition form factor for the Roper resonance transition is shown. Different model predictions are compared to previous data of the JLab-CLAS analysis and a new data point measured at MAMI-A1 for Q2 = 0.1 GeV2 [5]. Only the MAID prediction comes close to the new measurement because of the build-in constraint from the Siegert theorem. The study of baryon resonances is still an exciting field in hadron physics. With the partial wave analyses from MAID and the JLab group of electroproduc-tion data a series of transition form factors has been obtained in the space-like region. We have shown that with the help of the long-wavelength limit (Siegert Theorem) extrapolations to the time-like region can be obtained satisfying minimal constraints at the pseudo-threshold. In this time-like region, Dalitz decays in the process Nn —» N*/A —» Ne+e- can be measured and time-like form fac- Q 2 [GeV2] Fig. 3. Longitudinal transition form factor S-|/2(Q2) for the transition from the proton to the Roper resonance. The figure and the red exp. data point at Q2 = 0.1 GeV2 are from Stajner et al. [5], the blue data points are from the CLAS collaboration. The MAID model prediction which satisfies the Siegert's Theorem in the time-like region is in very good agreement with the new data point. For further details, see Ref. [5]. tors can be analyzed experimentally. Such experiments are already in progress at HADES@GSI and are also planned with the new FAIR facility at GSI. This work was supported by the Deutsche Forschungsgemeinschaft DFG (SFB 1044). References 1. L. Tiator, D. Drechsel, S. S. Kamalov and M. Vanderhaeghen, Eur. Phys. J. ST 198, 141 (2011). 2. I. G. Aznauryan and V. D. Burkert, Prog. Part. Nucl. Phys. 67,1 (2012). 3. D. Drechsel, S. S. Kamalov, and L. Tiator, Eur. Phys. J. A 34 (2007) 69; https: / / maid .kph.uni-mainz .de /. 4. R. A. Arndt, I. I. Strakovsky, R. L. Workman, Phys. Rev. C53 (1996) 430-440; (SP99 solution of the GW/SAID analysis); http://gwdac.phys.gwu.edu/. 5. S. Stajner et ni., Phys. Rev. Lett. 119, no. 2, 022001 (2017). Bled Workshops in Physics Vol. 18, No. 1 p. 57 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 Mathematical aspects of phase rotation ambiguities in partial wave analyses Y. Wunderlich Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, Nussallee 14-16, 53115 Bonn, Germany Abstract. The observables in a single-channel 2-body scattering problem remain invariant once the amplitude is multiplied by an overall energy- and angle-dependent phase. This invariance is known as the continuum ambiguity. Also, mostly in truncated partial wave analyses (TPWAs), discrete ambiguities originating from complex conjugation of roots are known to occur. In this note, it is shown that the general continuum ambiguity mixes partial waves and that for scalar particles, discrete ambiguities are just a subset of continuum ambiguities with a specific phase. A numerical method is outlined briefly, which can determine the relevant connecting phases. 1 Introduction We assume the well-known partial wave decomposition of the amplitude A(W, 0) for a 2 —> 2-scattering process of spinless particles A (W,0)= + 1)At (W)P^(cos 0). (1) «=o The data out of which partial waves shall be extracted are given by the differential cross section, which is (ignoring phase-space factors) °o (W,0) = |A (W, 0)|2 . (2) Making a complete experiment analysis [1] for this simple example, we see that the cross section constrains the amplitude to a circle for each energy and angle: |A(W, 0)| = +Voo(W, 0). Thus, one energy- and angle-dependent phase is in principle unknown when based on data alone. The other side of the medal in this case is given by the fact that the amplitude itself can be rotated by an arbitrary energy- and angle-dependent phase and the cross section does not change. This invariance is known as the continuum ambiguity [2]: A(W,0) -> A(W, 0) := ei®(W'e)A(W,0). (3) Another concept known in the literature on partial wave analyses is that of so-called discrete ambiguities [2-4]. Suppose the full amplitude A(W, 0) can be split into a product of a linear-factor of the angular variable, for instance cos 0, and a remainder-amplitude A(W, 0) [3]: A(W,0)= A(W,0)(cos 0 - a). (4) This is generally the case whenever the amplitude is a polynomial (i.e. the series (1) is truncated), but it may also be possible for infinite partial wave models. Then, it is seen quickly that the cross section (2) is invariant under complex conjugation of the root a, which causes the discrete ambiguity a —> a*. (5) Figure 1 shows a schematic illustration of the meaning of the terms continuum-vs. discrete ambiguities. In this proceeding, the purely mathematical mechanisms (3) and (5) are investigated. Of course, constraints from physics may reduce the amount of ambiguity encountered. For instance, unitarity is a very powerful constraint which, for elastic scatterings, leaves only one remaining non-trivial so-called Crichton-ambiguity [5]. This is believed to be true independent of any truncation-order L of the partial wave expansion [2]. However, in energy-regimes where the scattering becomes inelastic, so-called islands of ambiguity are known to exist [6]. X X X X Fig. 1. Three schematic pictures are shown in order to distinguish the terms discrete- and continuum ambiguities. The grey colored box depicts in each case the higher-dimensional parameter-space composed by the partial wave amplitudes, be it for infinite partial wave models, or for truncated ones. Left: One-dimensional (for instance circular) arcs can be traced out by continuum ambiguity transformations, both for infinite and truncated models. Center: Connected continua in amplitude space, containing an infinite number of points with identical cross section, can be generated by use of angle-dependent rotations (3) (however, only in case the partial wave series goes to infinity). The connected patches are also called islands of ambiguity [2,6]. Right: Discrete ambiguities refer to cases where the cross section is the same for discretely located points in amplitude space. These ambiguities are most prominent in TPWAs [2,4]. However, two-fold discrete ambiguities can also appear for infinite partial wave models, once elastic unitarity is valid [2]. These figures have been published in reference [8]. Although here we focus just on the scalar example, ambiguities have become a topic of interest in the quest for so-called complete experiments in reactions with spin, for instance photoproduction of pseudoscalar mesons [1,7]. This proceeding is a briefer version of the more detailed publication [8]. The arXiv-reference [9] also treats very similar issues, as does the contribution of Alfred Svarc to these proceedings. 2 The effect of continuum ambiguity transformations on partial wave decompositions We let the general transformation (3) act on A(W, 0) and assume a partial wave decomposition for the original as well as the rotated amplitude A(W, 0) —> A(W, 0) = ei®(W'e)A(W,0) = ei®(W'0^(2£ + 1 )A«(W)P«(cos 0) «=0 CO = + 1)A «(W)P«(cos 0). (6) «=0 Out of the infinitely many possibilities to parametrize the angular dependence of the phase-rotation, the convenient choice of a Legendre-series is employed co ei®(w,e) = £ Lk(W)Pk(cos 9). (7) k=0 In case this form of the rotation is inserted into the partial wave projection integrals of the general rotated waves At (cf. equation (6)), the following mixing formula emerges [10] k+« A«(W)=^ Lk(W) Y- (k,°;«,°|m,°)2 Am(W). (8) m=|k-«| Here, (ji, mi ; j2,m2|J, M) is just a usual Glebsch-Gordan coefficient. Some more remarks should be made on the formula (8): first of all, although it's derivation is not difficult, this author has (at least up to this point) not found this expression in the literature, at least in this particular form. However, mixing-phenomena have been pointed out for nN-scattering [11] and for photoproduction [12]. Secondly, in can be seen quickly from the mixing formula that for angle-independent phases, i.e. when only the coefficient L0 survives in the parametri-zation (7) of the rotation-functions, partial waves do not mix. Rather, in this case each partial wave is multiplied by L0 (W) = ei0(W). However, once the phase ®(W, 0) carries even a weak angle-dependence, the expansion (7) directly becomes infinite and thus introduces contributions to an infinite partial wave set via the mixing-formula. There may be (a lot of) cases where the series (7) converges quickly and in these instances, it is safe to truncate the infinite equation-system (8) at some point. It has to be stated that the mixing under very general continuum ambiguity transformations may lead to the mis-identification of resonance quantum numbers (reference [9] illustrates this fact on a toy-model example). 3 Discrete ambiguities as continuum ambiguity transformations In case of a polynomial-amplitude, i.e. a truncation of the infinite series (1) at some finite cutoff L, the amplitude decomposes into a product of linear factors [4] L L A(W, 9) = Y_ (2£ + 1) A« (W P« (cos 9) = A n (cos 9 - «0 , (9) «=0 i=1 with a complex normalization proportional to the highest wave A L. Furthermore, Gersten [4] claims (without proof) that the root-conjugations exhaust all possibilities for discrete ambiguities of the TPWA. We have to state that we believe him. The remainder of this proceeding is used to outline a numerical method that is orthogonal to the Gersten-formalism, but which can also substantiate this claim. 4 Functional minimizations show exhaustiveness of Gersten-ambiguities We use the notation x = cos 0, introduce the complex rotation function F(W, x) := ei®(w,x) and from now on drop the explicit energy W. The proposed numerical method assumes a truncated full amplitude A(x) as a known input. Then, all possible functions F(x) are scanned numerically for only those that satisfy the following two conditions: (I) The complex solution-function F(x) has to have modulus 1 for each value of x. |F(x)|2 = 1, Vx e [-1,1]. (13) (II) The rotated amplitude A(x), coming out of an amplitude A(x) truncated at L, has to be truncated as well, i.e. AL+k = 0, Vk = 1,..., oo. (14) Formally, this scanning-procedure can be implemented by minimizing a suitably defined functional of F(x): W [F(x)] := £ (Re [F(x)]2 + Im [F(x)]2 - 1) 2 + Im + Im r+1 dxF(x)A(x) + Z Re k>1 +1 dxF(x)A(x)PL+k(x) +1 2 dxF(x)A(x)PL+k(x) min. (15) Here, the first term ensures the unimodularity of F(x) (i.e. condition (I)), the second fixes a phase-convention on the S-wave Ao and the big sum over k sets all higher partial wave of the rotated amplitude to zero. It has to be clear that for practical numerical applications, the sums over k and x have to be finite, i.e. the former is cut off and the latter is defined on a grid of x-values. Also, a general function F(x) is defined by an infinite amount of real degrees of freedom, which has to be made finite as well. This can be achieved for instance by using a finite Legendre-expansion, i.e. a truncated version of equation (7) (with possibly large cutoff Lcut), or by discretiz-ing F(x) on a finite grid of points {xn} e [-1,1]. More details on the numerical minimizations can be found in reference [8]. The only non-redundant solutions of this procedure are, in the end, the Gersten-rotation functions (12). Figures 2 to 5 illustrate this fact for the simple toy-model [8] (partial waves given in arbitrary units): 2 A(x) = Y_ (2£ + 1 )■A«P (x) = Ao + 3Ai Pt (x) + 5A2P2 (x) t=o 5 = 5 + 3(0.4 + 0.3i)x + 5(0.02 + 0.01i)(3x2 - 1). (16) 2 2 2 2 2 eWo(x) 1 \u\ TO U 1.0 -0.5 0.0 cos(0) Nmax=100 0.5 1.0 cos(0) Nmax=200 cos(0) Nmax=300 cos(0) Nmax=150 cos(0) Nmax= 250 -1.0 -0.5 0.0 cos(0) Nmax= 500 0.5 1.0 Nrn,,=5 Nmax=50 Fig. 2. (Color online) The convergence of the functional minimization procedure is illustrated in these plots. For the discrete ambiguities eiVo(x) and eWl(x) of the toy-model (16), two randomly drawn initial functions have been chosen from an applied ensemble of initial conditions in the search. These initial conditions then converged to these two respective Gersten-rotations. Results are shown for different values of the maximal number of iterations Nmax of the minimizer, as indicated. Numbers range from Nmax _ 5 up to Nmax _ 500. In all plots, the real- and imaginary parts of the precise Gersten-ambiguity are drawn as blue and red solid lines. The results of the functional minimizations up to Nmax are drawn as thick dashed lines, having the same color-coding for real- and imaginary parts (color online). These figures have already been published in reference [8]. gWl (x) cos (0) cos (0) Fig. 3. These plots are the continuation of Figure 2. The convergence of the numerical minimization of the functional (15) is shown for the phase eiV1 (x), which generates discrete ambiguities of the toy-model (16). eW2M This model is truncated at L = 2. Thus it has two roots (a, a2) and 2 2 = 4 Gersten-ambiguities. The latter are generated by four phase-rotation functions: ei^o(x) = 1, e1^1 (x), eiV2(x) and e1^3^. Figures 2 to 5 demonstrate the convergence-process of the functional minimization towards a particular Gerstenrotation, for very general initial functions. The fact that always one of the four Gersten-rotations is found is independent of the choice of the initial function. 5 Conclusions & Outlook We have seen that general continuum ambiguity transformations, as well as discrete Gersten-ambiguities, are in the end manifestations of the same thing: angle-dependent phase-rotations. Therefore, they both mix partial waves. The rotations belonging to the Gersten-symmetries have the following defining property: they are the only rotations which, if applied to an original truncated model, leave the truncation order L untouched. In order to demonstrate this fact, a (possibly) new numerical method has been outlined capable of determining all continuum ambiguity transformations satisfying pre-defined constraints. A possible further avenue of reserach may consist off the generalization of these findings to reactions with spin, for instance pseudoscalar meson photoproduction. Here, the massive amount of new polarization data gathered over the last years have renewed interest in questions of the uniqueness of partial wave decompositions. However, once one transitions to the case with spin, some open issues still exist, as have already been discussed during the workshop. Acknowledgments The author (again, as in 2015) wishes to thank the organizers for the hospitality, as well as for providing a very relaxed and friendly atmosphere during the workshop. This particular Bled-workshop takes a special place in this author's biography, since after 4 months of battle with a very bad knee-injury, the participation in the workshop marked one of the first careful steps back into the world. Furthermore, the wonderful nature and environment of Bled itself turned out to be instrumental on the way of healing. By making the room on the ground floor of the Villa Plemeljavailable, the organizers have provided a key to make participation possible at all, and the author wishes to express deep gratitude for that. The author's wife also wishes to thank the organizers for the possibility to stay in Bled, as well as the nice hikes she made with the other participant's spouses. In fact, one early morning she was very brave and made a balloon ride over the lake of Bled. This author decided to include one of her aerial photographs into the proceeding. This work was supported by the Deutsche Forschungsgemeinschaft within the SFB/TR16. References 1. R.L. Workman, L. Tiator, Y. Wunderlich, M. Döring, H. Haberzettl, Phys. Rev. C 95 no.1, 015206 (2017). 2. J. E. Bowcock and H. Burkhardt, Rep. Prog. Phys. 38,1099 (1975). 3. L.P. Kok., Ambiguities in Phase Shift Analysis, In *Delhi 1976, Conference On Few Body Dynamics*, Amsterdam 1976, 43-46. 4. A. Gersten, Nucl. Phys. B 12, p. 537 (1969). 5. J. H. Crichton, Nuovo Cimento, A 45, 256 (1966). 6. D. Atkinson, L. P. Kok, M. de Roo and P. W. Johnson, Nucl. Phys. B 77,109 (1974). 7. Y. Wunderlich, R. Beck and L. Tiator, Phys. Rev. C 89, no. 5, 055203 (2014). 8. Y. Wunderlich, A. Svarc, R. L. Workman, L. Tiator and R. Beck, arXiv:1708.06840 [nucl-th]. 9. A. Svarc, Y. Wunderlich, H. Osmanovic, M. HadZimehmedovic, R. Omerovic, J. Stahov, V. Kashevarov, K. Nikonov, M. Ostrick, L. Tiator, and R. Workman, arXiv:1706.03211 [nucl-th]. 10. One just has to use the re-expansion k+£ , x 2 Pk (x)P, (x) = f^™) (2m + 1)Pm (x) m=|k-£| k+£ ^0 0 0, = Y- 0;£,0|m,0)2 Pm(x). (17) m=|k-£| For the first equality, see the reference: W. J. Thompson, Angular Momentum, John Wiley & Sons (2008). The second equality uses a well-known relation between 3j-symbols and Clebsch-Gordan coefficients. 11. N. W. Dean and P. Lee, Phys. Rev. D 5, 2741 (1972). 12. A. S. Omelaenko, Sov. J. Nucl. Phys. 34,406 (1981). Bled Workshops in Physics Vol. 18, No. 1 p. 68 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 Recent Belle Results on Hadron Spectroscopy M. Bracko University of Maribor, Smetanova ulica 17, SI-2000 Maribor, Slovenia and Jozef Stefan Institute, Jamova cesta 39, SI-1000 Ljubljana, Slovenia Abstract. Recent results on hadron spectroscopy from the Belle experiment are reviewed in this contribution. Results are based on experimental data sample collected by the Belle detector, which was in operation between 1999 and 2010 at the KEKB asymmetric-energy e+ e- collider in the KEK laboratory in Tsukuba, Japan. As a result of the size and quality of collected Belle experimental data, new measurements are still being performed now, almost a decade after the end of the Belle detector operation. Results from recent Belle publications on hadron spectroscopy, selected for this review, are within the scope of this workshop and reflect the interests of the participants. 1 Introduction During its operation between 1999 and 2010, the Belle detector [1] at the asymmetric-energy e+e- collider KEKB [2] accumulated an impressive sample of data, corresponding to more than 1 ab-1 of integrated luminosity. The KEKB collider, called a B Factory, was operating mostly around the Y(4S) resonance, but also at other Y resonances, like Y(1S), Y(2S), Y(5S) and Y(6S), as well as in the nearby continuum [3]. With succesful accelerator operation and excellent detector performance, the collected experimental data sample was suitable for various measurements, including the ones in hadron spectroscopy, like discoveries of new charmonium(-like) and bottomonium(-like) hadronic states, together with studies of their properties. 2 Charmonium and Charmonium-like states Around the year 2000, when the two B Factories started their operation [4], the charmonium spectroscopy was a well established field: the experimental spectrum of cc states below the DD threshold was in good agreement with theoretical prediction (see e.g. ref. [5]), and the last remaining cc states below the open-charm threshold were soon to be discovered [6]. 2.1 The X(3872)-related news However, the field experienced a true renaissance by discoveries of the so-called "XYZ" states—new charmonium-like states outside of the conventional charmo- nium picture. This fascinating story began in 2003, when Belle collaboration reported on B+ —} analysis1, where a new state decaying to was discovered [7]. The new state, called X(3872), was confirmed by the CDF, D0, BABAR collaborations [8], and later also by the LHC experiments [9]. The properties of this narrow state (r = (3.0+1.4 ± 0.9) MeV) with a mass of (3872.2 ± 0.8) MeV, which is very close to the D°D*° threshold [10], have been intensively studied by Belle and other experiments [11]. These studies determined the JPC = 1++ assignment, and suggested that the X(3872) state is a mixture of the conventional 23P1 cc state and a loosely bound D°D*° molecular state. If one wants to better understand the structure of X(3872), further studies of production and decay modes for this narrow exotic state are necessary. A recent example of these experimental studies at Belle is the search for X(3872) production via the B° -> X(3872)K+n- and B+ -> X(3872)K^n+ decay modes, where X(3872) decays to n- [12]. The results, obtained on a data sample contain- ing 772 x 106 BB events, show that B° -> X(3872) K* (892)° does not dominate the B° —} X(3872)(K+n-) decay, which is in clear contrast to charmonium behaviour in the B —» ^(2S)Kn case. Another consequence of the D°Dmolecular hypothesis of X(3872) is an existence of "X(3872)-like" molecular states with different quantum numbers. Searches for some of these states were performed in another recent Belle analysis [13], using final states containing the nc meson. A state X1 (3872), a D°D— ID°D*° combination with JPC = 1+-, and two states with JPC = 0++, X(3730) (combination of D°D° + D°D°) and X(4014) (combination of D*°D+ D*°D*°), were searched for. Additionally, neutral partners of the Z(3900)± [14] and Z(4020)± [15], and a poorly understood state X(3915) were also included in the search. No signal was observed in B decays to selected final states with the nc meson for any of these exotic states, so only 90% confidence-level upper limits were set. The interpretation of X(3872) being an admixture state of a D°Dmolecule and a xc1 (2P) charmonium state was also compatible with results of the recent Belle study of multi-body B decay modes with xc1 and xc2 in the final state, using the full Belle data sample of 772 x 106 BB events [16]. This study is important to understand the detailed dynamics of B meson decays, but at the same time these decays could be exploited to search for charmonium and charmonium-like exotic states in one of the intermediate final states such as xcjn and xcjnn. These recent results were already obtained with the complete Belle data sample, so more information about the nature of mentioned exotic states could only be extracted from the larger data sample, which will be available at the Belle II experiment [17]. 2.2 Alternative XcO (2P) candidate The charmonium-like state X(3915) was observed by the Belle Collaboration in B —» J/^^K decays [18]; originally it was named Y(3940). Subsequently, it was 1 Throughout the document, charge-conjugated modes are included in all decays, unless explicitly stated otherwise. also observed by the BABAR Collaboration in the same B decay mode [19] and by both Belle and BABAR in the process yy —> X(3915) —» J/^^ [20]. The quantum numbers of the X(3915) were measured to be JPC = 0++, and as a result, the X(3915) was identified as the Xco(2P) in the 2014 PDG tables [21]. However, many properties of the X(3915) state were found to be inconsistent with this identification. For example, the xc0 (2P) —> DD decay mode is expected to be dominant, but has not yet been observed experimentally for the X(3915). Also, the measured X(3915) width of (20 ± 5) MeV is much smaller than expected xc0(2P) width of r > 100 MeV [22]. A later reanalysis [23] of the data from Ref. [20] showed that both JPC = 0++ and 2++ assignments are possible. As a result of these considerations, the X(3915) was no longer identified as the Xc0 (2P) in the 2016 PDG tables [10]; and there was enough motivation for Belle Collaboration to perform an updated analysis of the process e+e- —» J/^DD. This latest analysis [24] used the 980 fb-1 data sample, collected at or near the Y(1S), Y(2S), Y(3S), Y(4S) and Y(5S) resonances. In addition to this 1.4 times increased statistics with respect to previous measurement, a sophisticated multi-variate method was used to improve the discrimination of the signal and background events, and an amplitude analysis was performed to study the JPC quantum numbers of the D D system. As a result of this analysis, a new charmonium-like state, the X* (3860), was observed in the process e+e- —» J/^DD. The mass of this state is determined to be (3862+26+43) MeV and its width is (201 +674+lD MeV. The X* (3860) quantum number hypotheses JPC = 0++ and 2++ are compared using MC simulation. Monte Carlo pseudoexperiments are generated according to the fit result with the 2++ X* (3860) signal in data and then fitted with the 2++ and 0++ signals (see Figure 1). The JPC = 0++ hypothesis is favoured over the 2++ hypothesis at the level of 2.5c. The new state X*(3860) seems to be a better candidate for the xc0(2P) char-monium state than the X(3915): the measured X*(3860) mass is close to potential model prediction for the xc0(2P), while the preferred quantum numbers are JPC = 0++, although the 2++ hypothesis is not excluded. 2.3 Study of JPC = 1 — states using ISR Initial-state radiation (ISR) has proven to be a powerful tool to search for JPC = 1 states at B-factories, since it allows one to scan a broad energy range of a/s below the initial e+e- centre-of-mass (CM) energy, while the high luminosity compensates for the suppression due to the hard-photon emission. Three charmonium-like 1 -- states were discovered at B factories via initial-state radiation in the last decade: the Y(4260) in e+e- -> J/^n+n- [25,26], and the Y(4360) and Y(4660) in e+e- —} ^(2S)n+n- [27,28]. Together with the conventional charmonium states ^(4040), ^(4160), and ^(4415), there are altogether six vector states; only five of these states are predicted in the mass region above the DD threshold by the potential models [29]. It is thus very likely, that some of these states are not charmo-nia, but have exotic nature—they could be multiquark states, meson molecules, quark-gluon hybrids, or some other structures. In order to understand the structure and behaviour of these states, it is therefore necessary to study them in many decay channels and with largest possible data samples available. a(-2 In L) Fig. 1. Comparison of the 0++ and 2++ hypotheses in the default model (constant nonreso-nant amplitude). The histograms are distributions of A(-2 ln L) in MC pseudoexperiments generated in accordance with the fit results with 2++ (open histogram) and 0++ (hatched histogram) signals. Recent paper from Belle collaboration [30] reports on the experimental study of the process e+e- —» yxcj (J=1, 2) via initial-state radiation using the data sample of 980 fb-1, collected at and around the Y(nS) (n=1, 2, 3, 4, 5) resonances. For the CM energy between 3.80 and 5.56 GeV, no significant e+e- —» YXc1 and yxc2 signals were observed except from ^(2S) decays, therefore only upper limits on the cross sections were determined at the 90% credibility level. Reported upper limits in this CM-energy interval range from few pb to a few tens of pb. Upper limits on the decay rate of the vector charmonium [^(4040), ^(4160), and ^(4415)] and charmonium-like [Y(4260), Y(4360), and Y(4660)] states to YXcJ were also reported in this study (see Table 1). The obtained results could help in better understanding the nature and properties of studied vector states. Xci (eV) Xc2 (eV) Fee[^(4040)] x YXcj] 2.9 4.6 Fee[^(4160)] x ß[^(4160) ^ YXcj] 2.2 6.1 Fee[^(4415)] x ß[^(4415) ^ YXcj] 0.47 2.3 Fee[Y(4260)] x B[Y(4260) ^ YXcj] 1.4 4.0 Fee[Y(4360)] x B[Y(4360) —> YXcj] 0.57 1.9 Fee[Y(4660)] x B[Y(4660) —> YXcj] 0.45 2.1 Table 1. Upper limits on Fee x B(R -> YXcj) at the 90% C.L. Initial-state radiation technique was also used in the new Belle measurement of the exclusive e+e- —» D(*'±D*+ cross sections as a function of the center-of-mass energy from the D(*'±D*+ threshold through y/s = 6.0 GeV [31]. The analysis is based on a Belle data sample collected with an integrated luminosity of 951 fb-1. The accuracy of the cross section measurement is increased by a factor of two over the previous Belle study, due to the larger data set, the improved track reconstruction, and the additional modes used in the D and D* reconstruction. The complex shape of the e+e- —» D*+D*- cross sections can be explained by the fact that its components can interfere constructively or destructively. The fit of this cross section is not trivial, because it must take into account the threshold and coupled-channels effects. Finally, the first angular analysis of the e+ e- —» D*±D*T process was performed within this study, allowing the decomposition of the corresponding exclusive cross section into three possible components for the longitudinally, and transversely-polarized D*± mesons, as shown in Figure 2. The obtained components have distinct behaviour near the D*+D*- threshold. The only non-vanishing component at higher energy is the TL helicity of the D*+D*- final state. The measured decomposition allows the future measurement of the couplings of vector charmonium states into different helicity components, useful in identifying their nature and in testing the heavy-quark symmetry. Is GeV Fig.2. The components of the e+e- —> D*+D*-yISR cross section corresponding to the different D*±'s helicities. (The labels and units for the horizontal axis, common in all three cases, are shown only for the right plot. 3 Results on Charmed Baryons Recently, a lot of effort in Belle has been put into studies of charmed baryons. Many of these analyses are still ongoing, but some of the results are already available. One example of such a result is the first observation of the decay A+ —» pK+n- using a 980 fb-1 data sample [32]. This is the first doubly Cabibbo-sup-pressed (DCS) decay of a charmed baryon to be observed, with statistical significance of 9.4 d (fit results for invariant-mass distributions are shown in Figure 3). The branching fraction of this decay with respect to its Cabibbo-favoured (CF) counterpart is measured to be B(A+ —» pK+n-)/B(A+ —» pK-n+) = (2.35 ± 0.27 ± 0.21) x 10-3, where the uncertainties are statistical and systematic, respectively. This year the results of the most recent baryon study were published [33]. in this study the inclusive production cross sections of hyperons and charmed 11 M(pKV) [GeV/c2] M(pK+n) [GeV/c2] Fig. 3. Invariant mass distributions for the A+ candidates: M(pK-n+) for the CF decay mode (left) and M(pK+n-) for the DCS decay mode (right, top). In the DCS case the distribution after the combinatorial-background subtraction is also shown (right, bottom). The curves indicate the fit result: the full fit model (solid) and the combinatorial background only (dashed). 400 OJ ! 300 LU 200 baryons from e+ e- annihilation were measured. The analysed sample corresponds to 800 fb-1 of Belle data collected around the Y(4S) resonance. The feed-down contributions from heavy particles were estimated and subtracted, using the measured data. The direct production cross sestions of hyperons and charmed baryons were thus measured and presented for the first time (see Figure 4). The production cross sections divided by the spin multiplicities for S = — 1 hyperons follow an exponential function with a single slope parameter except for the 1(1385) + resonance. A suppression for 1(1385) + and S = —2, —3 hyperons is observed, which is likely a consequence of decuplet suppression and strangeness suppression in the fragmentation process. The production cross sections of charmed baryons are significantly higher than those of excited hyperons, and strong suppression of Ic with respect to A+ is observed. The ratio of the production cross sections of A+ and Ic is consistent with the difference of the production probabilities of spin-0 and spin-1 diquarks in the fragmentation process. This observation supports the theory that the diquark production is the main process of charmed baryon production from e+e- annihilation, and that the diquark structure exists in the ground state and low-lying excited states of A+ baryons. 4 Summary and Conclusions Many new particles have already been discovered during the operation of the Belle experiment at the KEKB collider, and some of them are mentioned in this report. Although the operation of the experiment finished almost a decade ago, mass (GeV) mass (GeV) Fig. 4. Scaled direct production cross section as a function of mass of hyperons (left) and charmed baryons (right). S = — 1, —2, —3 hyperons are shown with filled circles, open circles and a triangle, respectively. data analyses are still ongoing and consequently more interesting results on char-monium(-like), bottomonium(-like) and baryon spectroscopy can still be expected from Belle in the near future. The results are eagerly awaited by the community and will be widely discussed at various occasions, in particular at workshops and conferences. Still, the era of the Belle experiment is slowly coming to an end. Further progress towards high-precision measurements—with possible experimental surprises — in the field of hadron spectroscopy are expected from the huge experimental data sample, which will be collected in the future by the Belle II experiment [17]. This future might actually start soon, since the Belle II detector begins its operation early next year. References 1. Belle Collaboration, Nucl. lustrum. Methods A 479,117 (2002). 2. S. Kurokawa and E. Kikutani, Nucl. lustrum. Methods A 499,1 (2003), and other papers included in this Volume. 3. J. Brodzicka et al., Prog. Theor. Exp. Phys., 04D001 (2012). 4. A. J. Bevan et al, Eur. Phys. J. C 74, 3026 (2014). 5. M. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008). 6. Belle Collaboration, Phys. Rev. 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Tomaradze and K. K. Seth, Phys. Lett. B 727, 366 (2013). 15. BESIII Collaboration, Phys. Rev. Lett. 111, 242001 (2013); Phys. Rev. Lett. 112, 132001 (2014). 16. Belle Collaboration, Phys. Rev. D 93, 052016 (2016). 17. Belle II Collaboration, Belle II Technical design report, [arXiv:1011.0352 [physics.ins-det]]. 18. K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 94,182002 (2005). 19. B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 101, 082001 (2008); P. del Amo Sanchez et al. (BaBar Collaboration), Phys. Rev. D 82, 011101 (2010). 20. S. Uehara et al. (Belle Collaboration), Phys. Rev. Lett. 104, 092001 (2010); J. P. Lees et al. (BaBar Collaboration), Phys. Rev. D 86, 072002 (2012). 21. K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014). 22. F. K. Guo and U. G. Meissner, Phys. Rev. D 86, 091501 (2012). 23. Z. Y. Zhou, Z. Xiao and H. Q. Zhou, Phys. Rev. Lett. 115, 022001 (2015). 24. K. Chilikin et al. (Belle Collaboration), Phys. Rev. D 95,112003 (2017). 25. BaBar Collaboration, Phys. Rev. Lett. 95,142001 (2005); Phys. Rev. D 86, 051102 (2012). 26. Belle Collaboration, Phys. Rev. Lett. 99,182004 (2007). 27. Belle Collaboration, Phys. Rev. Lett. 99,142002 (2007). 28. BaBar Collaboration, Phys. Rev. Lett. 98, 212001 (2007); Phys. Rev. D 89,111103 (2014). 29. S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985); T. Barnes, S. Godfrey and E. S. Swanson,Phys. Rev. D 72, 054026 (2005); G. J. Ding, J. J. Zhu and M. L. Yan, Phys. Rev. D 77, 014033 (2008). 30. Belle Collaboration, Phys. Rev. D 92, 012011 (2015). 31. Belle Collaboration, arXiv:1707.09167 [hep-ex]; submitted to Phys. Rev. D. 32. Belle Collaboration, Phys. Rev. Lett. 117, 011801 (2016). 33. Belle Collaboration, arXiv:1706.06791 [hep-ex]; submitted to Phys. Rev. D. Bled Workshops in Physics Vol. 18, No. 1 p. 76 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 The Roper resonance - a genuine three quark or a dynamically generated resonance? Faculty of Education, University of Ljubljana and Jozef Stefan Institute, 1000 Ljubljana, Abstract. We investigate two mechanisms for the formation of the Roper resonance: the excitation of a valence quark to the 2s state versus the dynamically generation of a quasi-bound meson-nucleon state. We use a coupled channel approach including the nN, nA and oN channels, fixing the pion-baryon vertices in the underlying quark model and using a phenomenological form for the s-wave sigma-baryon interaction. The LippmannSchwinger equation for the K matrix with a separable kernel is solved to all orders which results in the emergence of a quasi-bound state at around 1.4 GeV. Analysing the poles in the complex energy plane using the Laurent-Pietarinen expansion we conclude that the mass of the resonance is determined by the dynamically generated state, but an admixture of the (1s)2 (2s)1 component is crucial to reproduce the experimental width and the modulus of the resonance pole. This work has been done in collaboration with Simon Sirca from Ljubljana, Hedim Osmanovic from Tuzla and Alfred Svarc from Zagreb. The recent results of lattice QCD simulation in the P11 partial wave by the Graz-Ljubljana group [1] including besides 3q interpolating fields also operators for nN in relative p-wave and oN in s-wave, has revived the interest in the nature of the Roper resonance. Their calculation and a similar calculation by the Adelaide group [2] show no evidence for a dominant 3q configuration below 1.65 GeV and 2.0 GeV, respectively, that could be interpreted as a three-quark Roper state, and therefore support the dynamical origin of the Roper resonance. In our work [3] we study the interplay of the dynamically generated state and the three-quark resonant state in a simplified model incorporating the nN, nA and oN channels. The choice of the channels as well as of the parameters of the model is based on our previous calculations of the scattering and the meson photo- and electro-production amplitudes for several partial waves in which all relevant channels as well as most of the nucleon and A resonances in the intermediate energy regime have been included [5-9]. The bare octet-meson-baryon vertices are calculated in the Cloudy Bag Model while the parameters of the o-baryon interaction are left free: apart of its strength, the Breit-Wigner mass and the width of the o are varied. We have been able to consistently reproduce the results in the S and P partial waves; only the D waves typically require an increase in the strength of the meson-quark couplings compared to those predicted by the underlying quark model. The results presented here are obtained with the B. Golli Slovenia (j mass and width both equal to 600 MeV, and only the ctNN coupling is varied. Very similar results have been obtained for the mass and width of 500 MeV. The central quantity in our approach is the half-on-shell K matrix1 that consists of the resonant (pole) terms and the background (non-pole) term D: X (k k, ) = V-YNÎM VqN(k) + V-y^lCy) VqR(k) + p mN - W mR - W ay ^ kY) . (1) Indices a, ß, y ... denote the three channels, the first term corresponds to the nucleon pole, the second term is optional and generates an explicit resonance with the K-matrix pole at W = mR. The Lippmann-Schwinger equation (LSE) for the K matrix splits into the equation for the dressed N —» a vertex, VaN(k)= VON (k)+ £ dk ' K«ß(k,k ;)VßN (k/) Wß(k ')+ Eß (k ')- W' (2) and the equation for the background, Das(k,k5) = Kas(k,ks) + Y ß dk Kaß(k,k ;)PßS(k/,ks) ^ß(k')+ Eß(k')- W (3) If the resonant state is included, an equation analogous to (2) holds for the R —» a vertex. Let us note that the splitting of the K matrix is similar to the splitting used in approaches computing directly the T matrix, but is not equivalent. In the K-matrix approach the T matrix is obtained by solving the Heitler equation, T = K + iKT, which necessarily mixes the pole and the non-pole terms. Our approximation consists of assuming a separable form for the kernel Kap: Kaß(k,k')= £ ^ßi(k) ^ßi(k') , (4) vOi(k)=m1 (^ß+£ß«) Viß(k) ^a(k) + efß f1 aß • &(k ' ) = Via (k ' ) ^ß(k ' ) + efa 2 2 2 ß = mt- ma - ^ß 2E a where i runs over intermediate N and A, f are the corresponding spin-isospin factors, Vip corresponds to the decay of the baryon in channel |3 into the intermediate baryon and the meson in channel a, and m (E) and ^ stand for the baryon and the meson mass (energy), respectively. Kap(k, k') reduces to the u-channel exchange potential when either k or k ' takes its on-shell value. This type of approximation has been used in our previous calculations and has lead to consistent results. Let us mention that neglecting the integral terms in (2) and (3) corresponds to the so called K-matrix approximation. 1 x is proportional to the K matrix (satisfying S = (1 + iK) /(1 — iK)) by a kinematical factor which is not relevant for the present discussion. 78 B. Golli Equation (2) and (3) can be solved exactly by the ansatz: VaN (k) = yON (k) + £ x^i 9 (k), (5) '«Nl Pi _a5 (k) = (k, k5 ) + z«f 9 Pi (k), (6) Pi with coefficients x and z satisfying sets of algebraic equations of the form y ap xp = bP v ap zP5 = cP5 _ ai,Yj Yj ai , A._ ai,Yj ZYj ai Yj Yj Note that both equations involve the same matrix A = I + M, M = [M]Pt. where MP. . = ai,Yj dk &(k)9Pj(k) (7) d ^p(k)+ Ep (k) — W () For sufficiently strong interaction, the matrix A becomes singular and one or more poles appear in the background part of the K matrix which signals the emergence of a dynamically generated state. In fact, poles at the same energies appear also in the corresponding resonant terms of the K matrix, in addition to the nucleon pole and the (optional) pole at mR. The mechanism of this process can be studied by performing the singular value decomposition A = UWVT where W is a diagonal matrix containing the singular values wt. The singular values remain close to unity with exception of one which approaches zero as the interaction increases (Fig. 1 a) and eventually becomes negative for sufficiently strong gCTNN (Fig. 2 a). We claim that it is this value, wmin, and the corresponding singular vector Umin, that determine the properties of the quasi-bound molecular state. This state is dominated by the ctN component. For the invariant energies W for which wmin is close to zero, the solutions (5) and (6), in the absence of the resonant state R, can be cast in the form VaN (ka) « VON (ka) + — , Da6 (ka,k6) «Ka6(ka,k6) + . (8) wmin wmin Similarly, the nucleon self energy acquires the form v- r n VPN(k)yP°N(k) , Aw b In(W] = V_ dk-' PN « (mN — WW IN(W) + — J ^p (k) + EP(k)— W V N wm Just above the nN threshold, the D term is dominated by the u-channel N exchange processes which is reflected in a large peak in ImT (the non-pole term in Fig. 1 b). This term has the opposite sign with respect to the nucleon-pole term; these two terms almost cancel each other. In the energy region where wmin reaches its minimum the second terms in (8) and (9) dominate and the leading contribution to the K matrix reads K _ a«as_1__ ao b (mN — W) wmin wmin The two terms generate a resonance peak at the minimum of wmin (dashed-dotted line in Fig. 1 b); the real part, ReWp, of the corresponding S-matrix pole in Table 1 appears slightly below W of the minimum of wmin. Increasing goNN, wmin crosses zero twice and two poles of the S-matrix appear with ReWp close to the intersections (see Fig. 1 a and Table 1 for goNN = 2.05). If we include the resonant state by imposing a fixed value for mR in the second term of (1), the position of the peak almost does not change for a value of mR as low as 1530 MeV (solid line in Fig. 1 b). The effect of the resonant state is reflected in the increased width of the resonance rather than in the change of its position. This general scenario does not change if we decrease gnNN in order to reproduce the experimental values of ReT and ImT (Fig. 2 b). While the peak in ImT moves to somewhat higher W, the position of the minimum of wmin as well as of the real part of the S-matrix pole stay almost at the same value (see Table 1). Also, varying the value of mR between 1520 MeV and 2000 MeV has almost no influence on the behaviour of the amplitudes and the position of the S-matrix pole. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.8 0.6 0.4 0.2 non-pole - - - - non-pole + Npole----- + Rpole(2000) +Rpole(1530) - ReT -0.2 1200 1400 1600 1800 2000 2200 2400 W [MeV] a) 1100 1200 1300 1600 1700 1400 1500 W [MeV] b) Fig. 1. (Color online) a) The six lowest singular eigenvalues of the A matrix for g0NN = 2.0. b) The real and imaginary parts of the T matrix calculated from the background (non-pole) term alone (dashed lines), from the background plus the nucleon pole term (dash-dotted lines), and from including the resonant state either at mR = 1530 MeV (solid lines), or at mR = 2000 MeV (short-dashed lines) for goNN = 2.0. 1 0 We can summarize the results obtained in our simplified model as follows: • The main mechanism for the Roper resonance formation is the dynamical generation through a quasi-bound meson-baryon state around W « 1400 MeV dominated by the ctN component. Its mass is rather insensitive to variations of the gnNN coupling. 0.3 0.2 0.1 9aNN = 1.55 9aNN = 1.80 yy 9aNN = 2.00/^X 9aNN = 2.05 1 0.8 0.6 0.4 0.2 0 -0.2 non-pole - - - - non-pole + Npole----- + Rpole(2000)...... +Rpole(1530) - 1200 1300 1400 1500 1600 W [MeV] 1100 1200 1300 1400 1500 1600 1700 W [MeV] Fig. 2. (Color online) a) The lowest singular value of the W matrix, wmin, for four values of gaNN. b) Same as Fig. 1 b, except for gaNN = 1.55. 0 Table 1. S-matrix pole position and modulus for the model without the resonant state (mR = oo), and the model with the resonant state for two values of the K-matrix pole mass. The PDG values are taken from [10]. gaNN mR ReWp —2ImWp |r| £ [MeV] [MeV] [MeV] PDG 1370 180 46 —90° 1.80 oo 1397 157 11.2 —78° 2.00 oo 1358 111 20.6 —81° 2.05 oo 1331 44 7.3 —62° 1438 147 18.6 —17° 2.00 oo 1342 285 18.8 —11° gnNA = 0 1.55 2000 1368 180 48.0 —87° 1.55 1530 1367 180 47.5 —86° • The real part of the S-matrix pole, ReWp, remains close to or slightly below the mass of the quasi-bound state and is almost insensitive to the presence of a three-quark resonant state, while the PDG value of the imaginary part, ImWp, is reproduced only if the three-quark resonant state is included. • The S-matrix pole emerges with ReWp close to the minimum of wm;n even if (positive) wmin stays relatively far from zero; in this case the corresponding pole is not present in the K matrix. • The mass of the quasi-bound molecular state is most strongly influenced by the ctN component and lies ~ 100 MeV below the nominal ctN threshold; removing the nA component has little influence on the mass (see gnNA _ 0 entry in Table 1). References 1. C. B. Lang, L. Leskovec, M. Padmanath, S. Prelovsek, Phys. Rev. D 95, 014510 (2017). 2. A. L. Kiratidis et al., Phys. Rev. D 95, 074507 (2017). 3. B. Golli, H. Osmanovic, S. Sirca, and A. Svarc, arXiv:1709.09025 [hep-ph.] 4. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, and S. Sirca, Eur. Phys. J. A 26, 99 (2005). 5. B. Golli and S. Sirca, Eur. Phys. J. A 38, 271 (2008). 6. B. Golli, S. Sirca, and M. Fiolhais, Eur. Phys. J. A 42,185 (2009). 7. B. Golli, S. Sirca, Eur. Phys. J. A 47, 61 (2011). 8. B. Golli, S. Sirca, Eur. Phys. J. A 49,111 (2013). 9. B. Golli, S. Sirca, Eur. Phys. J. A 52, 279 (2016). 10. C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016) and 2017 update. Bled Workshops in Physics Vol. 18, No. 1 p. 82 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 Possibilities of detecting the DD* dimesons at Belle2 Mitja Rosina Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box 2964, 1001 Ljubljana, Slovenia and J. Stefan Institute, 1000 Ljubljana, Slovenia Abstract. The double charm dimeson DD* represents a very interesting four-body problem since it is a delicate superposition of a molecular (dimeson) and an atomic (tetraquark) configuration. It is expected to be either weakly bound or a low resonance, depending on the model. Therefore it is a sensitive test how similar are the effective quark-quark interactions between heavy quarks and light quarks. After the discovery of the E+ = ccd baryon at LHCb, there is a revived interest for the search of the double charm dimesons. There is, however, no such clear production and decay process available as it was for E+. Therefore we argue that it is, compared to LHCb, a better chance for the discovery of the DD* dimeson at the upgraded Belle-2 at KEK (Tsukuba, Japan) after 2019. 1 Introduction While the BB* dimeson (tetraquark) is expected to be strongly bound (>100 MeV) due to the smaller kinetic energy of the heavy quarks, the DD* dimeson is expected to be weakly bound (possibly at ~2 MeV) or a low resonance, depending on the model. Therefore it is a sensitive test of the effective quark-quark and quark-antiquark interactions. For example, can we assume Vuu = Vcu = Vcc = Vcû (apart from mass dependence of spin-dependent terms)? There is no such clear production and detection process available for the DD* intermediate state as it was for E++ which was recently discovered at LHCb analysing the resonant decay to A+K-n + where the A+ baryon was reconstructed in the decay mode pK-n+. Therefore we have started a study which production mechanism could enable the discovery of the DD* dimeson at the upgraded Belle-2 at KEK (Tsukuba, Japan) after 2019. For the time being, we summarize our old calculations of the DD* binding energy [1] and explain several tricky features of this interesting four-body system. 2 Comparison of charmed dimesons with the hydrogen molecule It is interesting to compare the molecule of two heavy (charmed) mesons with the hydrogen molecule. At short distance, the two protons in the hydrogen molecule are repelled by the electrostatic interaction, while the two heavy (charm) quarks in the mesonic molecule are attracted by the chromodynamic interaction because they can recouple their colour charges. Dimeson D+D* Molecule H+H Fig. 1. Difference between atom-like and molecular configurations Fig. 2. 3 Is the D+D* dimeson bound? In the restricted 4-body space assuming "cc" in a bound diquark state and the u and d quarks in a general wavefunction, the energy is above the D+D* threshold. In the restricted "molecular" 4-body space with the two c quarks far apart and a general wavefunction of u and d (as assumed by several authors), the energy is also above the D+D* threshold. Only combining both spaces (we took a rich 4-body space) brings the energy below the threshold. We should verify whether it happens also for other interactions (we have used the one-gluon exchange+linear confinement [1]). We failed to calculate the energy of the hidden charm (charmonium-like) meson X(3872) using the same method and interaction as for DD* [2]. The reason is that a perfect variational calculation in a rather complete 4-body space finds the absolute minimum of energy which corresponds to J/psi+eta rather than DD*. A demanding coupled channel calculation would be needed for a reliable result, and we have postponed it. It is an interesting question whether in the first step "cc" diquark is formed and later automatically dressed by u or d or u and d , or is the first step to form D + D* which merge into DD*. The later choice can profit from resonance formation, but due to the dense environment it is a danger that the D + D* system would again dissociate before really forming the dimeson. We intend to see which formalism would be appropriate for this. The dressing (fragmentation) of the b quark b — B- = bu 0.375+0.015 B° = ba 0.375+0.015 Bs = bs 0.160+0.025 Ab = bud 0.090+0.028 Fermilab CDF 2000 The dressing (fragmentation) of the cc diquark cc —- — ++ -cc = CCU 37% + — cc = ccd 37% Qcc+ = CCS 16% T + 1 cc =ccQd 9 % Fig. 3. The estimated probability of formation of the atomic tetraquark configuration compared to the Scc production Once the "cc" diquark is formed, it is probably dressed with one light quark into the Ecc baryon and only with about 9% probability into the "atomic" (cc)ud configuration. We have estimated this probability by analogy with the dressing of the b quark [3] into the Ab baryon compared to the production of B mesons (fig. 3). This percentage is further reduced by the evolution of the "atomic" configuration (cc)ud into the "molecular" configuration of DD*. 4 The decay of the DD* dimeson The DD* dimeson is stable against a two-body decay into D+D due to its quantum numbers I=0, J=1. It can decay, however, strongly in D+D+n, or electromagneti-cally in D+D+y, via the decay of D*. The strong decay is very slow (comparable to the electromagnetic decay) due to the extremely small phase space for the pion. Therefore, the DD* dimeson is "almost stable" and very suitable for detection. We are looking for convenient methods of detection. One possibility is related to the small phase space of the pionic decay [1] (fig. 4). The ratio between the pionic and gamma decay will strongly depend on the binding or resonance energy of the dimeson. Alternative suggestions are needed in order to have a reliable signature or tagging. We encourage the reader to come forth with new ideas! r> qql_._I_._I_._I_._I_,_i_,_ 13.94 13.95 13.96 13.97 13.98 13.99 m212 (GeV2) Fig. 4. Dalitz plot for the DD* decay depending on the binding or resonance energy; the area of the contours is proportional to the decay probability into pion 5 Conclusion Considering a rather large production cross section of double cc pairs at Belle, we expect a sufficient production rate of cc diquarks which get dressed by a light quark into a Ecc baryon. Once this expectation is verified, it is promising to search for the DD* dimesons, especially if they proceed via cc + u + d —> (cu)(cd). The motivation is twofold. • Since the DD* dimeson is a delicate system, it is barely bound or barely unbound, it would distinguish between different models. • Its production rate might help to understand the mechanism of the high production rate of double cc pairs at Belle. Work is in progress to study different production and decay mechanisms in order to find a tell-tale signature in the decay products. References 1. D. Janc and M. Rosina, Few-Body Systems 35 (2004) 175-196; also available at arXiv:hep-ph/0405208v2. 2. D. Janc, Bled Workshops in Physics 6, No. 1 (2005) 92; also available at http://www-f1.ijs.si/BledPub. 3. T. Affolder et al. (CDF Collaboration), Phys. Rev. Lett. 84 (2000) 1663. Bled Workshops in Physics Vol. 18, No. 1 p. 87 Proceedings of the Mini-Workshop Advances in Hadronic Resonances Bled, Slovenia, July 2 - 9, 2017 The study of the Roper resonance in double-polarized pion electroproduction a Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,1000 Ljubljana, Slovenia b JoZef Stefan Institute, Jamova 39,1000 Ljubljana, Slovenia investigations of the structure of the Roper resonance by using coincident electron scattering have been presented at several previous Mini-Workshops, and the most recent result on double-polarized pion electroproduction in the energy region of the Roper has recently been published [1]. This extended abstract is therefore just a reminder of the basic features of this experiment and just lists the highlights of that paper. Our experimental study of the p(e, e'p)n0 process was performed at the three spectrometer facility of the A1 Collaboration at the Mainz Microtron (MAMI) The kinematic ranges covered by our experiment were W « (1440 ± 40) MeV for the invariant mass, 9p « (90 ± 15)° and « (0 ± 30)° for the CM scattering angles and Q2 « (0.1 ± 0.02)(GeV/c)2 for the square of the four-momentum transfer. We have extracted the two helicity-dependent recoil polarization components, P^ and P^, as well as the helicity-independent component Py, and compared them to the values calculated by the state-of-the-art models MAID [2], DMT [3] and the partial-wave analysis SAID [4]. With the possible exception of Py at high W which is reproduced by neither of the models, MAID is in very good agreement with the data, while DMT underestimates all three polarization components and even misses the sign of P^. The SAID analysis agrees less well with the PX data, while it exhibits an opposite trend in Py and is completely at odds regarding P^. This might be a consequence of very different databases used in the analysis and calls for further investigations within these groups. We were also able to determine the scalar helicity amplitude S^2 in a model-dependent manner. In contrast to its transverse counterpart, A^2, this amplitude is accessible only by electroproduction (Q2 = 0) and becomes increasingly difficult to extract at small Q2. This is a highly relevant kinematic region where many proposed explanations of the structure of the Roper resonance and mechanisms of its excitation give completely different predictions. This is also a region in which large pion-cloud effects are anticipated. In the most relevant region below Q2 « 0.5 (GeV/c)2 where quark-core dominance is expected to give way to S. Sircaa'b manifestations of the pion cloud — and where existing data cease — the predictions deviate dramatically. Given that the agreement of our new recoil polarization data with the MAID model is quite satisfactory and that the transverse helicity amplitude Ai/2 is relatively much better known, we have performed a Monte Carlo simulation across the experimental acceptance to vary the relative strength of S with respect to the best MAID value for Ai/2 and made a x2-like analysis with respect to our experimentally extracted PX, Py and P^, of which Py was the most convenient for the fit. Fixing A^2 to its MAID value and taking SM/2ID as the nominal best model value, we have been able to express Si/2 from our fit as the fraction of Sm/2ID, yielding Si/2 _ (0.80-0:10) SM2ID _ (14.1-2:5) ■ 10-3Ge-i/2 . This result is shown in Fig. 3 of Loather Tiator's contribution to these Proceedings. References 1. S. Stajner et al., Phys. Rev. Lett. 119 (2017) 022001. 2. D. Drechsel, S. S. Kamalov, and L. Tiator, Eur. Phys. J. A 34 (2007) 69. 3. G. Y. Chen, S. S. Kamalov, S. N. Yang, D. Drechsel, and L. Tiator, Phys. Rev. C 76 (2007) 035206. 4. R. A. Arndt, W. J. Briscoe, M. W. Paris, and I. I. S. R. L. Workman, Chin. Phys. C 33 (2009) 1063. Povzetki v slovenščini Fotoprodukcija mezonov n in n' z modelom EtaMAID upoštevajoč Reggejevo fenomenologijo Viktor L. Kashevarov, Lothar Tiator, in Michael Ostrick Institut für Kernphysik, Johannes Gutenberg-Universitat, D-55099 Mainz, Germany Predstavimo novo verzijo modela EtaMAID za fotoprodukcijo mezonov p in p' na nukleonih. Model vsebuje 23 nukleonskih resonanc, ki jih opišemo z obliko Bre-ita in Wignerja. Ozadje opisemo z izmenjavo vektorskih in aksialno-vektorskih mezonov v kanalu t upostevajoc fenomenologijo Reggejevega reza. Parametri resonanc so bili prilagojeni znanim eksperimentalnim podatkom za fotoproduk-cijo mezonov p in p' na protonih in nevtronih. Razpravljamo o naravi najzanimivejših zapazanj. Vloga nukleonske resonance pri asimetriji nevtronov, ki se gibljejo izrazito naprej pri trkih visokoenergijskih polariziranih protonov na jedrih Itaru Nakagawa za kolaboracijo PHENIX RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Odkrili smo presenetljivo mocno odvisnost od mase pri enojni spinski asimetriji nevtronov, ki se gibljejo izrazito naprejpri trkih precno polariziranih protonov na jedrih pri energiji 200 GeV pri eksperimentu PHENIX na pospesevalniku RHIC. Taksna drastična odvisnost prekasa vsa pričakovanja običajnih hadronskih in-terakcijskih modelov. Odvisnost asimetrije od mase smo skusali teoreticno raz-loziti v okviru ultra perifernih trkov (efekt Primakoffa) z unitarnim izobarnim modelom (Mainz - MAID 2007). Racuni dajo dobro ujemanje. Racune z elektromagnetno interakcijo potrjuje slika, skladna z znanimi asimetrijskimi rezultati pri procesu pT + Pb —» n0 + p + Pb v Fermilabu. Analiza delnih valov pri fotoprodukciji mezonov n pri dani • •• •1 a •• • vi • • • 1*1* energiji - ilustracija z namišljenimi podatki H. Osmanovič1'*, M. HadZimehmedovič1, R. Omerovič1, S. Smajič1, J. Stahov1, V. Kashevarov2, K. Nikonov2, M. Ostričk2, L. Tiator2 and A. Svarč3 1 University of Tuzla, Fačulty of Natural Sčienčes and Mathematičs, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina 2 Institut fur Kernphysik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany 3 Rudjer Boskovič Institute, Bijenička česta 54, P.O. Box 180,10002 Zagreb, Croatia Z iterativnim postopkom kombiniramo analizo amplitud pri določenem t s kon-venčionalno analizo delnih valov pri določeni energiji na tak način, da rezultat ene analize sluzi kot omejitev pri drugi. Delovanje nase metode prikažemo na dobro definirani popolni zbirki namisljenih podatkov, ki smo jih proizvedli v okviru modela EtaMAID15. Ločljivost gruc pri relativističnih problemih malo teles Nikita Reičhelt1, Wolfgang Sčhweiger1 in William H. Klink2 1 Institute of Physičs, University of Graz, A-8010 Graz, Austria, 2 Department of Physičs and Astronomy, University of Iowa, Iowa čity, USA Relativistična kvantna mehanika je prikladen okvir za obravnavo zgradbe in dinamike hadronov v območju energijveč GeV. Drugače kot pri relativistični kvantni teorija polja zadosča tu določeno, ali vsajomejeno, stevilo prostostnih stopenj, da zagotovimo relativistično invariančo. Za sistem sodelujočih delčev to dosezzemo s tako imenovano Bakamjian-Tomasovo konstrukčijo, ki sistematsko vgradi inter-akčijske člene v generatorje Poinčarejeve grupe, tako da se ohranja njihova algebra. Ta metoda pa se sooči s fizično zahtevo ločljivost gruč, čim imamo več kot dvs delča. Ločljivost gruč, včasih jo imenujejo tudi "makroskopska kavzalnost", pomeni, da se ločena podsistema na dovoljveliki razdalji obnasata avtonomno. V tem prispevku razpravljamo o tem problemu in nakazemo resitev. Analiza delnih valov pri fotoprodukciji mezonov n pri dani energiji - eksperimentalni podatki J. Stahov1'*, H. Osmanovič1'*, M. HadZimehmedovič1, R. Omerovič1, V. Kashevarov2, K. Nikonov2, M. Ostričk2, L. Tiator2 and A. Svarč3 1 University of Tuzla, Fačulty of Natural Sčienčes and Mathematičs, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina 2 Institut fur Kernphysik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany 3 Rudjer Boskovič Institute, Bijenička česta 54, P.O. Box 180,10002 Zagreb, Croatia Analiza delnih valov pri fotoprodukciji mezonov n brez omejitev, v enem kanalu, pri eni energiji vodi do nezveznosti v energiji. Zveznost od točke do točke dosežemo z zahtevo po analitičosti pri fiksnem t na modelsko neodvisen način z uporabo razpolozljivih eksperimentalnih podatkov in pokazemo, da dosedanja baza podatkov ne zadosča za enolično resitev. Analitičost pri fiksnem t pri analizi amplitud s fiksnim t zagotovimo z metodo razvoja Pietarinena, ki je znana iz analize sipanja piona na nukleonu (Karlsruhe - Helsinki). Predstavimo analizo delnih valov z analitično omejitvijo za eksperimentalne podatke za stiri observable, ki so jih nedavno merili na pospesevalnikih MAMI in GRAAL v energijskem območju od praga do a/S = 1.85 GeV. Ekskluzivna fotoprodukcija pionov na vezanih nevtronih Igor Strakovsky The George Washington University, Washington, USA Podan je bil pregled dejavnosti skupine GW SAID pri analizi fotoprodukčije pionov na nevtronski tarči. Razvozlanje izoskalarnih in izovektorskih elektromagnetnih sklopitev resonanč N* in A* zahteva sprejemljive podatke na obojnih, pro-tonskih in nevtronskih tarčah. Interakčije med končnimi stanji igrajo kritično vlogo pri sodobni analizi reakčije yn —» nN na devteronski tarči. Resonančne sklopitve smo določili z metodo SAID PWA in jih primerjali s prejsnimi izsledki. Reakčije na nevtronih predstavljajo znaten delez studijv laboratorijih JLab, MAMI-C, SPring-8, ELSA in ELPH. Resonance in jakostne funkcije sistemov malo teles Yasuyuki Suzuki Department of Physičs, Niigata University, Niigata 950-2181, Japan and RIKEN Nishina Center, Wako 351-0198, Japan Resonanče nudijo preizkusni teren za dinamiko sistemov malo teles. Podrobno razpravljam o dveh tipih resonanč. Prva je ozka Hoylova resonanča v 12C, ki igra bistveno vlogo pri sintezi ogljika v zvezdah. Drugi tip pa so siroke, visoke resonanče z negativno parnostjo pri jedrih z masnim stevilom 4: 4H, 4He in 4Li. Pri prvem tipu je glavna čoulombska sila treh delčev alfa na velikih razdaljah, pri drugem tipu pa imamo jedrske sile kratkega dosega. Strukturoteh resonanč opisem z različnimi pristopi, in sičer z adiabatsko hipersferično metodo in kore-liranimi Gaussovimi funkčijam pri računih jakostnih funkčij. Resonanče uspesno lokaliziramo s kompleksnim absorpčijskim potenčialom, ozioma z metodo kompleksnega skaliranja. Od modela neodvisna pot od eksperimentalnih podatkov do parametrov polov (Večličnost kotne odvisnosti kontinuuma ter razvoj Laurenta in Pietarinena) Alfred Švarc Rudjer Boškovič Institute, Bijenička cesta 54, P.O. Box 180,10002 Zagreb, Croatia Kot je znano, da neomejena analiza delnih valov z eno energijo mnogo enakovrednih nezveznih rešitev, zato rabimo omejitev povezano s primernim teoretičnim modelom. Ce ne specificiramo kotne odvisnosti faze, ki povzroča večličnost kontinuuma, se mesajo multipoli; če pa izberemo fazo, resimo enoličnost resitve na modelsko neodvisen način. Doslejni bilo zanesljive metode, kako izvleči parametre polov iz tako dobljenih delnih valov, vendar smo pred kratkim razvili novo preprosto metodo z enim kanalom (razvojLaurenta in Pietarinena), kije uporabna tako za zvezne kot diskretne podatke. Uporabimo Laurentov razvojamplitude delnih valov, neresonantno ozadje pa razvijemo v potenčno vrsto za konformno preslikavo. Tako dobimo hitro konvergentno potenčno vrsto za preprosto analitično funkčijo z dobro definiranimi analitičnimi lastnostmi delnih valov, ki se ujemajo z vhodnimi podatki. Razvili smo tudi posplositev na več kanalov. Ce poenotimo obe metodi , lahko izpeljemo parametre polov neposredno iz eksperimentalnih podatkov brez skličevanja na katerikoli model. Prehodni oblikovni faktorji barionov od prostorskega pa do časovnega območja Lothar Tiator Institut für Kernphysik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany Predstavili smo razsiritev neelastičnih oblikovnih faktorjev za foto- in elektropro-dukčijo pionov na nukleonih iz območja negativnih kvadratov četvercev prenosa gibalne količine q2 v območje s pozitivnih Q2, vse tja do tako imenovanega psev-dopraga. V teh kinematičnih rezimih je mogoče določiti pomembne fizikalne omejitve za vijačnostne amplitude, ki sičer z neposredno meritvijo ne bi bile dostopne. š to metodo smo lahko nedavno določili tudi skalarno vijačnostno amplitudo Si/2 za Roperjevo resonančo. Matematične značilnosti večličnosti faznih zasukov pri analizah delnih valov Yannick Wunderlich Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, Germany Observable pri sipanju dveh teles v enem kanalu se ne spremenijo, če pomnoZimo amplitudo s skupno od energije in kota odvisno fazo. Ta invarianca je znana pod imenom večličnost kontinuuma. Poleg tega nastanejo znane diskretne veclicnosti zaradi kompleksne konjugacije korenov, zlasti pri okrnjeni analizi delnih valov. V tem prispevku pokazem, da splosna veclicnost kontinuuma mesa delne valove in da so za skalarne delce diskretne veclicnosti podmnozica kontinuumskih veclicnosti s specificno fazo. Na kratko orisem numericno metodo, ki lahko doloci ustrezne povezovalne faze. NovejSi rezultati spektroskopije hadronov pri eksperimentu Belle Marko Bracko Univerza v Mariboru, Smetanova ulica 17, 2000 Maribor, Slovenija in Institut Jozzef Stefan, Jamova cesta 39,1000 Ljubljana, Slovenija V tem prispevku so predstavljeni nekateri novejši rezultati spektroskopije hadro-nov pri eksperimentu Belle. Meritve so bile opravljene na vzorcu izmerjenih podatkov, ki ga je v casu svojega delovanja - med letoma 1999 in 2010 - zbral eksperiment Belle, postavljen ob trkalniku elektronov in pozitronov KEKB, ki je obratoval v laboratoriju KEK v Cukubi na Japonskem. Zaradi velikosti vzorca in kakovosti izmerjenih podatkov lahko raziskovalna skupina Belle se sedaj, ko je od zakljucka delovanja eksperimenta minilo ze skorajdesetletje, objavlja rezultate novih meritev. Izbor novejsih rezultatov, predstavljenih v tem prispevku, ustreza okviru delavnice in odraza zanimanje njenih udeležencev. Roperjeva resonanca - trikvarkovsko ali dinamično tvorjeno resonančno stanje? B. Golli Pedagoska fakulteta, Univerza v Ljubljani, Ljubljana, Slovenija in Institut J. Stefan, Ljubljana, Slovenija Raziskujemo dva mehanizma za tvorbo Roperjeve resonance: vzbuditev valen-cnega kvarka v orbitalo 2s v primerjavi z dinamicno tvorbo kvazivezanega stanja mezona in nukleona. Uporabimo pristop sklopljenih kanalov s tremi kanali nN, nA in oN, pri cemer dolocimo v kvarkovskem modelu pionska vozlisca z bar-ioni, za vozlisce z mezonom o pa vzamemo fenomenolosko obliko. Lippmann-Schwingerjevo enacbo s separabilnim jedrom za matriko K resimo v vseh redih, kar lahko vodi do nastanka kvazivezanega stanja v bliZini 1.4 GeV. Pole v kompleksni energijski ravnini analiziramo z Laurent-Pietarinenovim razvojem in ugotovimo, daje masa resonance določena z dinamično tvorjenjim stanjem, medtem ko je primes komponente (1s)2(2s)1 ključna za ujemanje z eksperimentalno določeno sirino resonanče in njenim modulom. Možnosti za odkritje dimezona DD* na detektorju Belle2 Mitja Rosina Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska 19, P.O.Box 2964,1001 Ljubljana, Slovenija in Institut Jozef Stefan, 1000 Ljubljana, Slovenija Dvojno čarobni dimezon DD* predstavlja zelo zanimiv problem stirih teles, ker je občutljiva superpozičija molekularne (dimezonske) in atomske (tetrakvarkovske) konfiguračije. Pričakujemo, da je bodisi sibko vezan, bodisi nizka resonanča, kar je odvisno od modela. Zato je občutljivo merilo, koliko so si podobne efektivne interakčije med tezkimi in lahkimi kvarki. Po odkritju bariona E+ = ččd na velikem hadronskem trkalniku LHCb v CERNu je ponovno zazivelo zanimanje za iskanje dvojno čarobnih dimezonov. Zal pa ni na voljo tako očitnih pročesov kot za produkčijo in razpad bariona E+. Zato predlagamo, da so boljši izgledi za odkritje dimezona DD* na povečanem detektorju Belle-2 v laboratoriju KEK v Tsukubi na Japonskem, ko bodo stekle meritve leta 2019. Študij Roperjeve resonance v dvojnopolarizirani elektroprodukciji pionov Simon Sirča Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska 19, P.O.Box 2964,1001 Ljubljana, Slovenija in Institut Jozef Stefan, 1000 Ljubljana, Slovenija Roperjeva resonanča in njena elektromagnetna struktura sodita med pomembne neresene uganke sodobne hadronske fizike. Lastnosti tega najnizjega vzbujenega stanja nukleona z istimi kvantnimi stevili so tezko dostopne, sajje resonanča skrita pod velikim ozadjem sosednjih resonanč. V prispevku smo poročali o meritvi polarizačijskih komponent odrinjenega protona iz pročesa p(e, e'p)n0, in sičer od vijačnosti odvisnih P¿, P¿ ter od vijačnosti neodvisne Py. Rezultate smo primerjali z modelskimi izračuni MAID, DMT in SAID ter ugotovili neujemanje zlasti pri slednjih dveh. Ob določenih modelskih privzetkih smo določili tudi skalarno vijačnostno amplitudo S1/2. Blejske Delavnice Iz Fizike, Letnik 18, št. 1, ISSN 1580-4992 Bled Workshops in Physics, Vol. 18, No. 1 Zbornik delavnice 'Napredek pri hadronskih resonancah', Bled, 2. - 9. julij 2017 Proceedings of the Mini-Workshop 'Advances in Hadronic Resonances', Bled, July 2-9, 2017 Uredili in oblikovali Bojan Golli, Mitja Rosina, Simon Sirca Članki so recenzirani. Recenzijo je opravil uredniski odbor. Izid publikacije je finančno podprla Javna agencija za raziskovalno dejavnost RS iz sredstev drzavnega proracuna iz naslova razpisa za sofinanciranje domacih znanstvenih periodicnih publikacij. Tehnicni urednik Matjaz Zaversnik Zalozilo: DMFA - zaloznistvo, Jadranska 19,1000 Ljubljana, Slovenija Natisnila tiskarna Itagraf v nakladi 80 izvodov Publikacija DMFA stevilka 2051 Brezplacni izvod za udelezence delavnice