Blejske delavnice iz fizike Letnik 16, št. 1 Bled Workshops in Physics Vol. 16, No. 1 ISSN 1580-4992 Proceedings of the Mini-Workshop Exploring Hadron Resonances Bled, Slovenia, July 5 -11, 2015 Edited by Bojan Golli Mitja Rosina Simon Sirca University of Ljubljana and Jozef Stefan Institute dmfa - ZALOŽNIŠTVO Ljubljana, november 2015 The Mini-Workshop Exploring Hadron 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 Joief Stefan Institute, Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Organizing Committee Mitja Rosina, Bojan Golli, Simon Sirca List of participants Marko Braiko, Ljubljana, marko.bracko@ijs.si Ido Gilary, Haifa, chgilary@tx.technion.ac.il Bojan Golli, Ljubljana, bojan.golli@ijs.si Ju-Hyun Jung, Graz, ju.jung@uni-graz.at Viktor Kashevarov, Mainz, kashev@kph.uni-mainz.de Dubravko Klabuiar, Zagreb, klabucar@oberon.phy.hr Luka Leskovec, Ljubljana, luka.leskovec@ijs.si Felipe Llanes-Estrada, Madrid, fllanes@ucm.es Willi Plessas, Graz, willibald.plessas@uni-graz.at Mitja Rosina, Ljubljana, mitja.rosina@ijs.si George Rupp, Lisboa, george@ist.utl.pt 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 Alfred Svarc, 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 Resonance states and branching ratios from a time-dependent perspective. Ido Gilary.......................................................... 1 In-medium properties of the nucleon within a n-p-^ model Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim...................... 4 n MAID-2015: update with new data and new resonances V. L. Kashevarov, L. Tiator, M. Ostrick.................................. 9 Analytic structure of nonperturbative quark propagators and meson processes Dalibor Kekez and Dubravko Klabucar................................. 16 Comparing mesons and WLWL TeV-resonances Felipe J. Llanes-Estrada etal........................................... 20 Resonances in the Constituent-Quark Model R. Kleinhappel and W. Plessas......................................... 27 Unquenched quark-model calculation of excited p resonances and P-wave nn phase shifts Susana Coito, George Rupp, Eef van Beveren............................ 30 The pion-cloud contribution to the electromagnetic nucleon structure D. Kupelwieser and W. Schweiger..................................... 36 Partial wave analysis of n photoproduction data with analyticity constraints J.Stahovetal........................................................ 40 IV Contents Progress in Neutron Couplings W. J. Briscoe and I. Strakovsky......................................... 49 Exciting Baryon Resonances with Meson Photoproduction L. Tiator, A. Svarc.................................................... 59 Complete experiments in pseudoscalar meson photoproduction Y. Wunderlich....................................................... 68 Recent Spectroscopy Results from Belle M. Bracko........................................................... 75 Eta and kaon production in a chiral quark model B.Golli............................................................. 81 Vector and scalar charmonium resonances with lattice QCD Luka Leskovec, C.B. Lang, Daniel Mohler, Sasa Prelovsek................. 87 Resonances in the Nambu-Jona-Lasinio model MitjaRosina ........................................................ 91 The Roper resonance — Ignoramus ignorabimus? S. Sirca............................................................. 96 Povzetki v slovenscini 101 Preface At the end of the day, it looked like a resonance. We came together, exchanged news and ideas, and separated. The decay time seemed to be one week, but the ideas kept on boiling until the tail (the Proceedings) has been reached after four months. We hope that this final state shall trigger new resonances or echoes in the next years at Bled again. The emphasis has been on the interpretation of experimental data and on the development of new theoretical and computational techniques for resonances. The prevailing experiments to generate and study low-energy hadronic resonances have been the photoproduction and electroproduction of mesons. The prevailing theoretical advance has been in exploiting analytic properties of scattering and production amplitudes. Light hadron spectroscopy remains an exciting field in nuclear and particle physics, The Roper resonance is still an elusive entity, however, new experiments at MAMI, Jefferson Lab and other laboratories have elucidated several attempts at explaining it within models involving quark+meson and baryon+meson degrees of freedom. The progress beyond the Roper has required very elaborate partial-wave analyses, strongly based on recently measured polarization observables, and disentanglement of the isoscalar and isovector electromagnetic couplings. Theoretical studies took account of analytical constraints and of the final-state interactions which play a critical role. The photoproduction and electroproduction of n, n' mesons and kaons has opened an avenue to a detailed study of several nonstrange and strange resonances. Nucleon structure is still a motivating topic, for example, the question of the pion-cloud contribution to the electromagnetic properties, or, the issue of medium modifications of the structure of nucleons bound in a nuclear medium. In the meson sector, radial recurrences of the p(770) vector resonance have been studied in terms of nn P-wave phase shifts, including all relevant decay channels. The resonances X(3872) and Z± (4430,4200,4050,4250) are now better understood. Charmonium resonances in D-D scattering, including ^(3770), have been studied in Lattice QCD, suggesting a narrow resonance below 4 GeV. A WLWL TeV-resonance has been speculated about, based on the analogy with meson-meson resonances and hints from LHC. We would like to thank all participants for coming and for making, once more, the Mini-Workshop so friendly, lively and fruitful. The aim of the Proceedings is to prevent the vivid impressions of talks and discussions from fading away. The problems that have been opened are now documented in order to stimulate our further mutual interactions. Ljubljana, November 2015 B. Golli, M. Rosina, S. Sirca Predgovor Doživeli smo resonanco! Zbrali smo se, izmenjali novice in ideje, in se spet razšli. Razpadni cas najbi bil en teden, toda nove ideje so vrele se vec mesecev in rep (Zbornik) so dosegle po stirih mesecih. Upamo, da bo ta resonanca sprožila nove resonance ali odmeve v prihodnjih letih na Bledu. Poudarek je bil na tolmačenju eksperimentalnih podatkov in na razvoju novih teoretičnih in racunskih vescin za resonance. Prevladujoci eksperimenti za tvorbo in raziskavo nizkoenergijskih hadronskih resonanc so bili fotoprodukcija in elek-troprodukcija mezonov. Prevladujoc napredek v teoriji pa je bil v izkoriscanju analiticnih lastnosti amplitud za sipanje in procese. Spektroskopija lahkih hadronov je se vedno vzpodbudno podrocje fizike jedra in delcev. Roperjeva resonanca je se vedno izmuzljiva, vendar so novi eksperimenti v laboratorijih MAMI, Jefferson Lab in drugih osvetlili razlage s prostost-nimi stopnjami kvark-mezon in barion-mezon. Napredek iznad Roperja je zahteval zelo prefinjeno analizo delnih valov, tudi za nedavno merjene polarizacijske kolicine, ter razlocitev izoskalarnih in izovektorskih elektromagnetnih sklopitev. Teoreticne obravnave pa so upostevale analiticne omejitve ter zelo pomembno interakcijo v koncnem stanju. Fotoprodukcija in elektroprodukcija mezonov r, r' in kaonov je odprla pot podrobnim raziskavam stevilnih navadnih in cudnih resonanc. Zgradba nukleona se gradi naprej. Zanimiv je prispevek pionskega oblaka k elektromagnetnim lastnostim nukleona. Posebno podrocje je raziskava vpliva okolice na zgradbo nukleona potopljenega v jedrsko snov. V mezonskem podrocju je bil uspesen studijradialnih ponovitev vektorske resonance p (770) v okviru faznih premikov za sipanje mezonov n + n v valu P in z vkljucitvijo vseh vaznih kanalov. Napredek je tudi pri razumevanju resonanc X(3872) in Z±(4430, 4200, 4050,4250). S kromodinamiko na mrezi so raziskovali carmonijske resonance v sipanju mezonov D-D (zlasti ^(3770)) in napovedali ozko resonanco nekoliko pod 4 GeV. Z nekajdomisljije (in namigom z Velikega hadronskega trkalnika) bi lahko napovedali resonanco WLWL pri eneriji vec TeV po analogiji z resonancami mezon-mezon. Radi bi se zahvalili vsem udeležencem, da so napravili naso mini-delavnico tako prijazno, zivahno in plodno. Namen pricujocega Zbornika je, najzivi vtisi pre-davanjin razprav ne splahnijo. Problemi, ki smo jih odprli, so zabelezeni z namenom, da vzpodbujajo nadaljnje sodelovanje med nami. Ljubljana, november 2015 B. Golli, M. Rosina, S. Sirca Workshops organized at Bled > What Comes beyond the Standard Model (June 29-July 9,1998), Vol. 0 (1999) No. 1 (July 22-31,1999) (july 17-31, 2000) (july 16-28, 2001), Vol. 2 (2001) No. 2 (july 14-25, 2002), Vol. 3 (2002) No. 4 (july 18-28, 2003) Vol. 4 (2003) Nos. 2-3 (july 19-31, 2004), Vol. 5 (2004) No. 2 (july 19-29, 2005), Vol. 6 (2005) No. 2 (September 16-26, 2006), Vol. 7 (2006) No. 2 (july 17-27, 2007), Vol. 8 (2007) No. 2 (july 15-25, 2008), Vol. 9 (2008) No. 2 (july 14-24, 2009), Vol. 10 (2009) No. 2 (july 12-22, 2010), Vol. 11 (2010) No. 2 (july 11-21, 2011), Vol. 12 (2011) No. 2 (july 9-19, 2012), Vol. 13 (2012) No. 2 (july 14-21, 2013), Vol. 14 (2013) No. 2 (july 20-28, 2014), Vol. 15 (2014) No. 2 (july 11-20, 2015), Vol. 16 (2015) No. 2 > Hadrons as Solitons (july 6-17,1999) > Few-Quark Problems (july 8-15, 2000), Vol. 1 (2000) No. 1 > Statistical Mechanics of Complex Systems (August 27-September 2, 2000) > Selected Few-Body Problems in Hadronic and Atomic Physics (july 7-14,2001), Vol. 2 (2001) No. 1 > Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25-28, 2001) > Quarks and Hadrons (july 6-13, 2002), Vol. 3 (2002) No. 3 > Effective Quark-Quark Interaction (july 7-14, 2003), Vol. 4 (2003) No. 1 > Quark Dynamics (july 12-19, 2004), Vol. 5 (2004) No. 1 > Exciting Hadrons (july 11-18, 2005), Vol. 6 (2005) No. 1 > Progress in Quark Models (july 10-17, 2006), Vol. 7 (2006) No. 1 > Hadron Structure and Lattice QCD (july 9-16, 2007), Vol. 8 (2007) No. 1 > Few-Quark States and the Continuum (September 15-22, 2008), Vol. 9 (2008) No. 1 > Problems in Multi-Quark States (june 29-july 6, 2009), Vol. 10 (2009) No. 1 > Dressing Hadrons (july 4-11, 2010), Vol. 11 (2010) No. 1 > Understanding hadronic spectra (july 3-10, 2011), Vol. 12 (2011) No. 1 > Hadronic Resonances (july 1-8, 2012), Vol. 13 (2012) No. 1 > Looking into Hadrons (july 7-14, 2013), Vol. 14 (2013) No. 1 > Quark Masses and Hadron Spectra (july 6-13, 2014), Vol. 15 (2014) No. 1 > Exploring Hadron Resonances (july 5-11, 2015), Vol. 16 (2015) No. 1 X Participants Satellite view (top) and the GPS track (bottom) of the two-boat rowing excursion on Lake Bled on July 11, 2015. Bled Workshops in Physics Vol. 16, No. 1 p.l A Proceedings of the Mini-Workshop Exploring Hadron Resonances Bled, Slovenia, July 5 - 11, 2015 Resonance states and branching ratios from a time-dependent perspective. Ido Gilary Shulich Faculty of Chemistry, Technion, Haifa, 3200003, Israel The spectrum of a given Hermitian quantum mechanical system can be generally separated into a discrete part containing the bound states and a continuous part of scattering states above the threshold. These states are solutions to the time-independent Schrodinger equation (TISE) with the corresponding boundary conditions. The discrete nature of the bound spectrum enables the characterization of the stable part of the studied system along with its physical properties based on its energy levels. The continuum can be used to characterize the system by probing it through scattering experiments. In this context instead of discrete energy levels the discussion is usually shifted to resonances in the scattering profile of the studied system. These appear as sharp features in the energy profile of the interaction. Wavepacket dynamics in the continuum of meta-stable open quantum systems reveals that in the interaction region the evolution resembles that of bound states. There is, however, one difference where a bound system preserves probability in a meta-stable one when we observe decay in time. When the decay from the interaction region tends to follow an exponential form then the energy content of the localized part of the wavepacket assumes a constant value. This value is complex where the real part represents the energy of a resonance of the system and the imaginary part is related to the width of this resonance. The dynamics outside the interaction region exhibit a spatial exponential increase which drops off at the edge of the escaping wavefront. The velocity of the escaping part of the wavepacket has the momentum corresponding to the average energy inside. Similar dynamics is observed when scattering a wavepacket off a potential at a resonant energy. Initially the arriving wavepacket populates a resonant boundlike state inside the interaction region and consequently the formed meta-stable state decays. The time-dependent dynamics observed in meta-stable systems demonstrates the properties of both stationary bound and scattering states. This type of dynamics usually occurs due to either the shape of the potential of interaction where the variation can lead to confinement of finite time or due the coupling of a bound state in an closed channel with the continuum of an open channel. The first type of states is often termed shape-type resonance whereas the second type is usually called Feshbach-type resonances. The above discussion suggests that the essence of the dynamics can be captured by solving a time independent equation with appropriate boundary conditions. Such boundary conditions allow only outgo- 2 Ido Gilary ing flux. Solving the TISE with outgoing boundary conditions leads to solutions with complex momentum. This makes the energies of these states complex just as the portrayed dynamics and it also displays the asymptotic divergence which was observed. In order to be able to calculate the energies of the resonance states one needs to be able to fix the asymptotic divergence. This can be achieved in various ways which all lead to a non-Hermitian Hamiltonian. Some of the techniques used are: (1) scaling of the coordinate by a complex factor (complex scaling); (2) addition of a complex absorbing potential at the asymptotes far from the interaction region; (3) Feshabch projection formalism which separates between the spaces of localized and scattering states; (4) Siegert pseudo states which satisfy the required boundary condition on a given surface but lead to a quadratic eiegnvalue problem. The use of Non-Hermitian Hamiltonians readily yields the information regarding the lifetime of a given meta-stable state in addition to its energy. On the other hand, it leads to some complications due the non-Hermiticity. First of all, the resonance states are not orthogonal with respect to the conventional scalar Dirac product. Instead one needs to find and additional set of states which are orthogonal to them. These are the eigenstates of the Hermitian conjugate Hamiltonian which physically are their time-reversed counterparts. The two biorthogonal states form together the resolution of the identity. Another aspect is the loss of the probabilistic interpretation due to the non-Hermiticty. This can be amended by redefining the inner product based on the bi-orthogonal set of states. By doing so the probabilistic interpretation is retained along with the non-unitary evolution resulting from the decay of the system. This allows to describe very complicated dynamics in the continuum based on dynamics of several resonance states alone. In decaying few-body systems there are often several channels open to decay. In such case the decay rate of the resonance contains contributions due to the flux in each of the open channels. When considering the above mentioned outgoing boundary conditions one finds that the momentum of the outgoing flux in each channel depends on the resonance energy and the threshold energy of the given channel. When following the wavepacket dynamics in such systems one observes that at every channel the wavepacket leaks at the velocity given by the momentum in that channel. Consequently all the information regarding the partial widths to the different channels and their branching ratios can be extracted from the stationary resonance wavefunction. All that is needed in order to evaluate the branching ratios is the complex amplitude of the resonance wavefunction at the asymptotes and the momentum at every given channel which is obtained from the difference between the resonance energy and the channel's threshold energy. Resonance states and branching ratios from a time-dependent perspective. 3 References 1. Shachar Klaiman and Ido Gilary, "On Resonance: A First Glance into the Behavior of Unstable States." In: Advances in Quantum Chemistry, 63. "Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications", Edited by Cleanthes A. Nicolaides and Erkki Brandas. San Diego: Academic Press, pp. 131 (2012). 2. Tamar Goldzak, Ido Gilary and Nimrod Moiseyev, "Evaluation of partial widths and branching ratios from resonance wave functions." Physical Review A. 82,052105 (2010). Bled Workshops in Physics Vol. 16, No. 1 p. 4 A Proceedings of the Mini-Workshop Exploring Hadron Resonances Bled, Slovenia, July 5 - 11, 2015 In-medium properties of the nucleon within a n-p-^ model* Ju-Hyun Junga, Ulugbek Yakhshievb, Hyun-Chul Kimb aTheoretical Physics, Institute of Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria b Department of Physics, Inha University, Incheon 402-751, Republic of Korea Abstract. In this talk, we report on a recent investigation of the transverse charge and energy-momentum densities of the nucleon in the nuclear medium, based on an in-medium modified n-p-w soliton model. The results allow us to establish general features of medium modifications of the structure of nucleons bound in a nuclear medium. We briefly discuss the results of the transverse charge and energy-momentum densities. 1 Introduction The generalized parton distributions (GPDs) provide a new aspect of the structure of the nucleon, since they contain essential information on how the constituents of the nucleon behave inside a nucleon. The energy-momentum tensor (EMT) form factors (FFs) are given by Mellin moments of certain GPDs and characterize how mass, spin, and internal forces are distributed inside a nucleon. The EMT FFs are essential quantities in understanding the internal structure of the nucleon [1-3]. Furthermore the transverse charge which is defined by a Fourier transform in the transverse plane provides a tomographic picture of how the charge densities of quarks are distributed transversely [4,5]. 2 Lagrangian of the model We start from the in-medium modified effective chiral Lagrangian with the n, p, and w meson degrees of freedom, where the nucleon arises as a topological soliton. Using the asteriks to indicate medium modified quantities, the Lagrangian has the form — + LV + Lkin + (1) * Talk delivered by Ju-Hyun Jung In-medium properties of the nucleon within a n-p-w model where the corresponding terms are expressed as f 2 f 2 f 2 m2 = Tr (3oU3oU^ - ^ ^ Tr (3^3^) + as^ f2 2 LV = f Tr [D,^ • ^ + • ^ , Tr (U - 1) , (2) (3) r> * _ Lkin — 2gV Zv Tr (F 2 ] J LWz H ^T Tr {(Ut3vU) (Ut3aU) (Ut3pU)} . (4) (5) Here, the SU(2) chiral field is written as U = £,[ £,R in unitary gauge, and the field-strength tensor and the covariant derivative are defined, respectively, as = 02Vv - SvV, - i[V2,Vv] D2 £l(R) = d2 ^L(R) - Î V2 ^L(R) . (6) (7) We assume the following ansatze for the pseudoscalar and vector mesons U = exp ^ — F(r) V, = g (t • p, + œ,), Po = 0, Pi" = g^ G(r), œ, = œ(r)5,0 (8) with the Pauli matrices t in isospin space. One can minimize the static mass functional related to the Lagrangian in Eq. (1) and find the solitonic solutions corresponding to a unit baryon number (B = 1 ). The integrand of the static mass functional corresponds to T00 component of the energy momentum tensor presented below. The details of the minimization procedure can be found in Ref. [6]. Using the Lagrangian in Eq. (1), one can calculate each component of the EMT as follows: T00* (r) = ap f| ( 2 ^ + F A + af^m* (1 - cos F) 2 r2 (1 - cos F + G)2 - Zg2f>2 +{2r2G '2 +G2 (g+2)2}-r 2g2Zr2 '3 1 T0i* (r, s) = gilmrlgm (s x r) 2n2r2 T PJ (r) , œsin2 FF', Tij* (r) = s (r) ( ^ - ^ ) + p (r) 5ij , r2 3 (9) (10) (11) 5 6 Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim where f2 1 n PÍ M = 3f 3A + 3g2r2ZA is the angular momentum density and sin2 F + 8 sin4 2 - 4 sin2 2^,1 1 [(2 - 2£,! - £,2) G2] + sin2 FF' (12) 1 p* (r)=-6apfn F'2 + 2 r2 sin2 f' asfnm7t (1 - cos F) 2 3Í2 fn (1 - cos F + G)2 + f^g2Z—2 +2r2 G '2 + G2 (g+2)2} + '2' 6g2 Zr2 s* (r)= f F'2 - - f (1 - cos F + G)2 r r2 (13) + g2r2Z [r2G'2 - G2 (G + 2)2} 2 (14) are the pressure and the shear force distributions inside the nucleon. The moment of inertia of the rotating soliton including 1/Nc corrections is given by the expression A* = 4n dr f (sin2 F + 8 sin4 2 - 4 sin2 2^ + 2 gVZ, {(2 - 2^ - ^2) G2} + ® sin2 FF' 3g2r2Z 4n2 (15) As we mentioned above, the integral of Too gives the soliton mass at zero momentum transfer t = 0. Therefore, the M2 (t) form factor is normalized by the nucleon mass as M2 (0) = 1 mn d3r Too (r) = 1 (16) to leading order in MN, which is equals to the soliton mass [7]. For details, we refer to Refs. [8,9] The EMT FFs of the nucleon parametrize the nucleon matrix elements of the symmetric EMT operator as follows [2,3]:
= ü(p', s')
M2(q2) + J(q2) + Pv^.p)qP (17)
Mn
2
+d!(q2) q.qv -
2Mn
5Mn
ü(p, s) ,
(18)
where P = (p + p ')/2.
In-medium properties of the nucleón within a n-p-œ model
7
One can be related GPDs. In the specific case, £ = 0, one has
r i
A20 (t) = M2 (t) =
-1
dxxH (x,0,t) ,
B20 (t) = 2J (t) - M2 (t) =
dxxE (x, 0, t) .
(19)
(20)
In the isospin symmetric approximation the proton and neutron EMT FFs are similar. Therefore we introduce the nucleon transverse EMT densities instead of considering the proton and neutron EMT densities separately. In this approximation an unpolarized nucleon transverse EMT density takes the form
p02) (b)=r (bQ) A2° (q2) •
For a polarized nucleon one has the following transverse EMT density
pT2)(b) = p°2) (b)- sin - fo)
(21)
Q2dQ .
4nMN
ji (bQ) B20 (Q2) .
(22)
x
3 Results
Now let us discuss the energy-momentum form factors of the nucleons. First of all, it is necessary to notice that in the case of exact isospin symmetry the energy-momentum form factors of the protons and neutrons cannot be distinguished in free space. The same result holds for the nucleons embedded in isospin symmetric nuclear matter. The situation changes if one introduces isospin breaking effects in the mesonic sector. In the case of in-medium nucleons the isospin asymmetric nuclear environment can generate differences in EMT form factors of the nucleons even if one has isospin symmetry in the mesonic sector in free space. For simplicity we concentrate in this work on the isopin symmetric case for both, free space as well as in medium nucleons, considering an isospin symmetric nuclear environment.
The energy-momentum form factors of the nucleons as functions of t are presented in Fig. 1 for free space nucleons and in-medium nucleons at normal nuclear matter density p°.
Finally, in Fig. 2 we present the transverse energy-momentum densities inside an unpolarized and polarized nucleon for the fixed value of bx = 0.
Our complete results will appear in some detail elsewhere [10].
Acknowledgments
This work is supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Korean government (Ministry of Education, Science and Technology), Grant No. 2011-0023478 (J.H.Jung and U.Yakhshiev) and Grant No. 2012004024 (H.Ch.Kim). J.-H.Jung. acknowledges also partial support by the "Fonds zur Forderung der wissenschaftlichen Forschung in Österreich via FWF DK W1203-N16."
8 Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim
Fig. 1. The EMT form factors of the nucleon, A20 and B20, as functions of t. The solid curve depicts the form factors in free space. The dotted and dotted-dashed ones represent, respectively, those from Model I and Model II in nuclear medium at the normal nuclear matter density p0.
Fig. 2. Transverse energy-momentum densities inside an unpolarized and polarized nucleon with bx = 0 fixed. The solid curve depicts the form factors in free space. The dotted and dotted-dashed ones represent, respectively, those from model I and model II in nuclear matter.
References
1. X. D. Ji, Phys. Rev. D 55, 7114 (1997).
2. X. D. Ji, Phys. Rev. Lett. 78, 610 (1997).
3. M. V. Polyakov, Phys. Lett. B 555, 57 (2003).
4. U. Yakhshiev and H. -Ch. Kim, Phys. Lett. B 726, 375 (2013).
5. K. Goeke, M. V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47 401 (2001).
6. J. -H. Jung, U. T. Yakhshiev and H. -Ch. Kim, Phys. Lett. B 723, 442 (2013).
7. H. -Ch. Kim, P. Schweitzer and U. T. Yakhshiev, Phys. Lett. B 718, 625 (2012).
8. J. -H. Jung, U. Yakhshiev and H. -Ch. Kim, J. Phys. G 41, 055107 (2014).
9. J. -H. Jung, U. Yakhshiev, H. -Ch. Kim and P. Schweitzer Phys. Rev. D 89,114021 (2014).
10. J. -H. Jung, U. T. Yakhshiev and H. -Ch. Kim, in preparation.
11. T. Ericson and W. Weise, Pions and Nuclei (Clarendon, Oxford, 1988).
Bled Workshops in Physics Vol. 16, No. 1 p. 9
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
n MAID-2015: update with new data and new resonances*
V. L. Kashevarov, L. Tiator, M. Ostrick
Institut fur Kernphysik, Johannes Gutenberg-Universitat, D-55099 Mainz, Germany
Abstract. Recent data for n and n photoproduction on protons obtained by the A2 Collaboration at MAMI are presented. The total cross section for n photoproduction demonstrates a cusp at the energy corresponding to the n threshold. The new data and existing data from GRAAL, CBELSA/TAPS, and CLAS collaborations have been analyzed by an expansion in terms of associated Legendre polynomials. The isobar model r|MAID updated with r|/ channel and new resonances have been used to fit the new data. The new solution r|MAID-2015 reasonably good describes the data in the photon beam energy region up to 3.7 GeV.
1 Introduction
The unitarity isobar model nMAID [1] was developed in 2002 for n photo- and electroproduction on nucleons. The model includes a nonresonant 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. The Born terms are evaluated with the pseudoscalar coupling. The vector meson contribution is obtained by the p and w meson exchange in the t channel with polelike Feynman propagators. For each partial wave the resonance contribution is parameterized by the Breit-Wigner function with energy dependent widths. The nMAID-2003 version includes eight resonances, N(1520)3/2-, N(1535)1/2-, N(1650)1/2-, N(1675)5/2-, N(1680)5/2+, N(1700)3/2-, N(1710)1/2+, N (1720)3/2+, and was fitted to proton data for differential cross sections and beam asymmetry at photon beam energies up to 1400 MeV. The nMAID-2003 version describes not only the experimental data available in 2002, but even a bump structure around W=1700 MeV in n photoproduction on the neutron, which was observed a few years later. However, this version fails to reproduce the new polarization data obtained in Mainz [2].
The aim of this work is to extend the nMAID-2003 version to higher energies, to improve a description of the new polarization data, and to include the n1 photoproduction channel.
* Talk presented by V. L. Kashevarov
10
V. L. Kashevarov, L. Tiator, M. Ostrick
2 Truncated Legendre analysis
The full angular coverage of differential cross sections and polarization observables allow us to perform a fit with a Legendre series truncated to a maximum orbital angular momentum £max:
do
2 i„
^ = £ A^Pn(cos©n),
do
n=0
21
T(F) âQ = Z An(F)Pn(cos©n),
I drQ = H AnPn(cos©n),
n= 1
2^max
dQ
(1) (2) (3)
n=2
where P^ (cos 0n) are the associated Legendre polynomials. The spin-dependent cross sections, Tda/dH, Fda/dH, and Ida/dO were obtained by multiplying the corresponding asymmetries with the differential cross sections obtained in Mainz. As an example, the results for the Legendre coefficients for differential
rñ
a?
1.5 2 2.5
1.5 2 2.5 1.5 2
W [GeV]
1.5 2 2.5
Fig. 1. Legendre coefficients in [|b/sr] up to £max = 5 from our fits to the differential cross section of the yp —> r|p reaction as function of the center-of-mass energy W. Red circles are fit results for preliminary A2/MAMI data [3], black and blue - for CBELSA/TAPS [4] and CLAS [5] data correspondingly.
0
U.05
0.05
0
cross sections are presented in Figs. 1 and 2. A non-zero Aio only posible with h-wave contribution, A9 is dominated by the an interference between g and h waves, A8 includes g, h waves and an interference between f and h waves, and so on. The first coefficient, A0, was omitted in the figures because of it includes all possible partial-wave amplitudes and just only reflects the magnitude of the total cross section, see Fig. 3.
Non-zero values of the A7 and A8 coefficients point to a contribution of the g wave at energies above W=2 GeV for both n and n channels. The errors in
n MAID-2015 11
0
0.04
0
-0.02 -0.04
if f\ .........h-
........[if.
-(
a a
H* -,-,-ji-,-. ........h
^0.04 -0.02
A7
ter M
2 2.2 2.4 2.6
2 2.2 2.4 2.6
aO
A3
ao A8
i-n^-
2 2.2 2.4 2.6
W [GeV]
a"
A4
t' r"' '
a"
a9
2 2.2 2.4 2.6
Fig. 2. The same as Fig. 1, but for the yp —> |p reaction.
a"
A5
a "
10
2 2.2 2.4 2.6
0.05
0.04
025
025
0
the determination of the coefficients A9 and Aio do not allow any conclusions about the contribution of h wave in these reactions. Polarization observables for r photoprduction were measured below W=1.9 GeV. The Legendre fit for these data shows the sensitivity to small partial-wave contributions and indicates pd interferences below W=1.6 GeV and df interferences above W=1.6 GeV [2].
3 Updated nMAID
New nMAID-2015 model is based on the |MAID-2003 version. The following main changes were made:
• 12 additional resonances were added: N (1860)5/2+, N(1875)3/2-, N (1880)1/2+, N (1895) 1/2-, N (1900)3/2+, N (1990)7/2+, N (2000)5/2+, N(2060)5/2-, N(2120)3/2-, N(2190)7/2-, N(2220)9/2+, and N(2250)9/2-;
• electromagnetic couplings for the vector mesons were updated according to Ref. [6];
• hadronic vector and tensor couplings for the vector mesons were fixed from Ref. [7];
• data base for the fit was updated.
The new model was fitted to data of differential cross sections from the A2 Collaboration at MAMI [3] and CLAS Collaboration [5], polarisation observables T, F [2] and I [8], [9]. The main variable parameters for each resonance: Breit-Wigner mass, total width, branching ratio to np (or rp) decay, photoexcitation helicity amplitudes A^2 and A3/2, a relative sign between the N* —» |N and the N* —} nN couplings. Besides, the hadronic pseudoscalar coupling for the Born term contribution, cutoffs for dipole formfactors of the vector mesons, damping factors for the partial widths and the electromagnetic form factor of the resonances were also fitted. Branching ratios for hadronic decays of the resonances besides the investigated channel were fixed.
As an initial parameter set for the Breit-Wigner parameters the last BnGa solution [10] was used. As initial parameter limits uncertainties from Refs. [6]
12
V. L. Kashevarov, L. Tiator, M. Ostrick
and [10] were used. As the first step, for each resonance A^2 and A3/2 are fixed because of a strong correlation with the branching ratio. On the second step the branching ratios obtained on the first step are fixed, but A^2 and A3/2 are variable, and so on. After few iterations the initial limits are changed if necessary. The fits for the n and n' channels were done independently.
The fit results for the total cross sections and the polarization observables are presented in Figs. 3-6 together with corresponding experimental data. We used the differential cross section from the CLAS Collabration [5] in this fit because of their much smaller statistical errors, larger energy covering, and better agreement with the high statistic data from A2/MAMI [3] in an overlapping energy region. Unfortunately, the total cross section was not determined in Ref. [5] and we calculated it using Legendre decomposition for the differential cross sections. Blue circles in Figs. 3 and 5 are results of this procedure.
In Fig. 3, there is a very interesting feature at energy -1900 MeV, which could be explained by a cusp due to the opening of a new channel, n' photoproduction. The main resonance, which is responsible for this effect is the N(1895)1/2-. The Breit-Wigner parameters of this state were determined by the fit as following: M = 1896 ± 1 MeV, rtot = 93 ± 13 MeV, rnp = (14 ± 3)%, F^ = (6.5 ± 2)%, and Ai/2 = (—17.4±1.5) 10-3 GeV-1/2. Fig. 4 demonstrates a significant improvement of description for T and F asymmetries (red lines) in comparison with the nMAID-2003 version (blue lines).
Fig.3. Total cross section of the Yp —> t|P reaction. Solid blue curve is r|MAID-2003 isobar model [1], black solid curve: new r|MAID-2015 solution. Prediction of r|MAID-2003 for background contribution is shown by blue dashed line, background of r|MAID-2015 -black dashed line. Vertical lines correspond to thresholds of KZ, œ, and n photoproductions.
n MAID-2015 13
W=1.497 GeV
W=1.646 GeV
-W=1-.-497GeVi 0 2
tU-l.
W=1.646 GeV i4!iii!í$
. w=i,5i6 Gev_.
--W=1:51^ GeV-i] 02
W=1.674 GEV
W=1.496 W=1
PS
W=1.754
zftlzfl
W=1.754-i 1 W=1.783
-W=1.5-34GfeV-
■»A
W=1.702 GeV
•iM" e
W=1.549
LI
05 :-W=T.5-58 GfeV-
W=1.743 GeV
0^--W=1".558-GeV-
1 ;-W=-t.743-GeV-
^ W=1.796 GEV il ^ W=1.848 GEV
0 5 r W=1.5S8- GeV-j 0 GfeV-
0' : : 1 :
1 W=1.796 GeV 1 W=1.848 GeV
W=1*619
.........
4..................... 0
^-W=1í/8eM 1
W=1.884
W=1.908
0 1-1 0 1-1 0 1-1 0 1-1 0 1-1 0 1
cos©„
Fig. 4. nMAID-2015 solution for the n channel (red lines). Black circles: A2/MAMI-15 data [2] for T and F asymmetries, blue circles: GRAAL-07 data [8] for I. Blue lines: nMAID-2003 prediction [1].
125
0 1
1 W=1.81
0 1
A very good agreement with the experimental data was obtained for the cross section of the YP —» n'P reaction (see Fig. 5). The main contributions to this reaction come from N ( 1895) 1/2-, N (1900)3/2+, N (1880) 1/2+, N (2150)3/2-, and N (2000)5/2+ resonances. Other resonance contributions are much smaller then the background. The new nMAID-2015 solution describes shape of the GRAAL data for I near threshold, but not the magnitude (see Fig. 6). To explain, why the magnitude of the asymmetry is larger at lower energy, it is probably necessary to include below threshold resonances using the more realistic approach applied in Ref. [11] for the Roper resonance at n-meson photoproduction.
4 Summary and conclusions
In summary, we have presented new version nMAID-2015. The model describes available data for the yp —» np and yp —» n'p reactions reasonably well. The cusp at W-1900 MeV in yp —» np reaction was explained as a threshold effect from the n' channel. Parameters of N(1895)1/2- resonance, responsible for this effect, were determined. A further improvement could be achieved by adding below threshold resonances and using Regge trajectories for the vector mesons in t channel. Furthermore, polarization observables which should come soon from A2/MAMI, CBELSA/TAPS, and CLAS Collaborations will help to improve the model.
14
V. L. Kashevarov, L. Tiator, M. Ostrick
Fig. 5. Total cross section of the yp —> r\!V reaction. Red circles: A2/MAMI-15 data [3], black circles: CBELSA/TAPS-09 [4], blue circles: data obtained from the Legendre fit to the differential cross sections of the CLAS Collaboration [5]. Solid black line: r|MAID-2015 solution. Background contribution is shown by dashed black line. Black dotted and dot-dashed lines are partial contributions of the Born terms and the vector mesons correspondingly. Other curves are partial contributions of resonances.
0.2 0.1 W 0 -0.1 -0.2 0.2 0.1 W 0 -0.1 -0.2
-1 -0.5 0 0.5 1 cos 0
Fig. 6. Beam asymmetry Z. Data from Ref. [9], red curves are r|MAID-2015 solution.
Acknowledgment
This work was supported by the Deutsche Forschungsgemeinschaft (SFB 1044).
------------ --W-1-.90 32 GeV 5 o
T —-—ir
J-i —-d t- 1
i ii --------------
------------- W-1.91 25 GeV --------------
I T^T £
-1- o -1- _L
------------ --------------
n MAID-2015 15
References
1. W. -T. Chiang, S. N. Yang, L. Tiator, and D. Drechsel, Nucl. Phys. A700, 429 (2002).
2. J. Akondi et al. (A2 Collaboration at MAMI), Phys. Rev. Lett. 113,102001 (2014).
3. P. Adlarson et al. (A2 Collaboration at MAMI), Submitted to Phys. Rev. Lett.
4. V. Crede et al. (CBELSA/TAPS Collaboration), Phys. Rev. C 80, 055202 (2009).
5. M. Williams et al. (CLAS Collaboration), Phys. Rev. C 80, 045213 (2009).
6. K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014).
7. J. M. Laget et al, Phys. Rev. C 72, 022202(R) (2005).
8. O. Bartalini et al. (The GRAAL Collaboration), Eur. Phys. J. A 33,169 (2007).
9. G. Mandaglio et al., EPJ Web of Conferences 72 00016 (2014); P. Levi Sandri et al., arXiv:1407.6991v2.
10. A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev, U. Thoma, Eur. Phys. J. A 48,15 (2012).
11. I. G. Aznauryan, Phys. Rev. C 68, 065204 (2003).
Bled Workshops in Physics Vol. 16, No. 1 p. 16
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Analytic structure of nonperturbative quark propagators and meson processes*
Dalibor Kekeza and Dubravko Klabucarb
a Rugjer Boskovic Institute, Bijenicka c. 54,10000 Zagreb, Croatia b Physics Department, Faculty of Science, Zagreb University, Bijenicka c. 32, Zagreb 10000, Croatia
Abstract. The analytic structure of certain Ansätze for quark propagators in the nonperturbative regime of QCD is investigated. When choosing physically motivated parameterization of the momentum-dependent dressed quark mass function M(p2), with definite analytic structure, it is highly nontrivial to predict and control the analytic structure of the corresponding nonperturbative quark propagator. The issue of the Wick rotation relating the Minkowski-space and Euclidean-space formulations is also highly nontrivial in the nonperturbative case. A propagator form allowing the Wick rotation and enabling equivalent calculations in Minkowski and Euclidean spaces is achieved. In spite of its simplicity, this model yields good qualitative and semi-quantitative description of some pseudoscalar meson processes.
Lattice studies of QCD are complemented by the continuum QCD studies utilizing Dyson-Schwinger equations (DSE). Both ab initio DSE studies and DSE studies for models of QCD provide an important approach for the study of phenomena in hadronic physics both at zero and finite temperatures and densities -see, for example, Refs. [1,2]. Just like lattice QCD studies, the large majority of DSE calculations (including those of our group, e.g., [3]) are implemented in the Euclidean metric.
Nevertheless, solutions of the Bethe-Salpeter equation require analytic continuation of DSE solutions for dressed quark propagators Sq (p), into the complex p2-plane. Similar situation is with the processes that involve quark propagators (QP) and Bethe-Salpeter amplitudes: it is not enough to know propagators and the Bethe-Salpeter amplitude only in the spacelike region, for real and positive p2. It is important to know the analytic properties in the whole p2 complex plane.
Alkofer et äl. [4] have explored the analytic structure of the Landau gauge gluon and quark propagators. They have proposed some simple analytic Ansätze for these propagators. Based on their Ansätze, Jiang et äl. [5] provide an analytical approach to calculating the pion decay constant fn and and the pion mass Mn at finite density.
We want to investigate and further improve the analytic structure of the quark propagator S(p). It can be conventionally parameterized (in Minkowski
* Talk delivered by Dubravko Klabucar
Analytic structure of nonperturbative quark propagators ... 17
space) as
s(p) = -M-P2) P + (-p2) = z(-p2)-
and correspondingly in Euclidean space as
,2w , „r p + M(-p2)
jp2 - M2(-p2)
?M -, ¡2^ ¡2, Z(p2) ^ ip + M(p2)
S (p) = ipCTv(p2) + CTs(p2) = . , , , ., ^ = Z(p2)-
-ip + M(p2) >2 + M2(p2) '
where M(x) is the dressed quark mass function and Z(x) is the wave function renormalization. Alkofer et al. [4] have explored the analytic structure of the quark (and gluon) propagator in the Landau gauge, using numerical solutions of the pertinent Dyson-Schwinger equations and fits to lattice data as inputs. Their Ansatze for Z and M (or crv and crs) include meromorphic functions (poles on the real axes or/and pairs of the complex conjugate poles) and functions with branch cut structures. Positivity violation in the spectral representation of the propagator shows the presence of the negative-norm contributions to the spectral function, i.e., the absence of asymptotic states from the physical part of the state space, which is sufficient (but not necessary) criterion for the confinement. While in the gluon propagator a clear evidence for positivity violation is found, the similar analysis shows that there is probably no such violation in the quark propagator [4]. The propagator with pairs of complex conjugate poles violates causality. It has been argued [6,7] that the corresponding S-matrix remains both causal and unitary (see also Ref. [1]).
Furthermore, complex conjugate poles can pose a problem for the analytic continuation from Minkowski to Euclidean space (Wick rotation) used by lattice gauge theory and functional methods. It has been also shown that complex conjugate poles in S(p) cause thermodynamical instabilities at nonvanishing temperature and density [8].
Of crucial importance is the following question: Is it possible to find an analytic Ansatz for the quark propagator solely with branching cut (or cuts) on the real timelike axes, with no additional structures (isolated singularities or cuts) in the complex plane? Such an Ansatz could be used for practical calculation of the processes involving quark loops.
Because of a complicated interplay between analytic structure of the functions Z and M on one side, and crs and crv on the other side, the approximation A = 1 has been applied. Then, the problem reduces to finding of appropriate functions M(x) and cr(x) = 1/(x + M2(x)). The most rigorous constraint is that the propagator S(p) —» 0 for all directions |p2| ^ oo in the complex p2 plane [9]. Furthermore, for large and positive values of x = p2 (spacelike momenta), function M(x) must be positive and approach to zero from above [4]. In the Euclidean regime, for real and positive values of x, the mass function should be fitted to match the form known from lattice and Dyson-Schwinger calculations.
Number of Ansatze for the quark mass function has been investigated. When choosing certain parametrization of the function M(x), with definite analytic structure, it is highly nontrivial to predict and control the analytic structure of the accompanying cr(x) function. Relevant mathematical theory and possibly related theorems (like Rouches theorem) are hardly applicable for this concrete problem.
18 Dalibor Kekez and Dubravko Klabucar
The best results were achieved with the Ansatz of the form M(x) = log(R(x)), where R is a rational function with certain good properties. The function M(x) has a few cuts on the real timelike axes, while the propagator dressing function ct(x) has both branch cuts and poles on the real timelike axes. No additional structure are present in the complex momentum plane. The quark propagator based on this Ansatz should allow for the Wick rotation and equivalent calculation in Minkowski and Euclidean spacetime.
Future work will include an improved fitting of the mass function M(x) and refinement of calculation with Z(x) = 1. Furthermore, we are planning to check whether our Ansatz satisfies the requirements of positivity violation.
The quark propagator obtained in this way, endowed with good analytic properties, should then be tried and adjusted so that it gives good results in various applications: the yy-transition and charge form factors of pions, ct and p form factors and decays, are just some of the interesting potential applications of the quark propagator Ansatz with good analytic structure. It is also necessary to investigate the related issue of the Bethe-Salpeter equation in Minkowski space. The quark loop contribution to various processes should also be studied using these improved quark propagators. Besides the processes like n,n,n' -;> YY that are described by an anomalous triangle diagram, there are interesting anomalous processes based on the pentagon diagram, like n,n' —^> 4n. (We could expect new results from high-statistics n' experiments like BES-III, ELSA, CB-at-MAMIC, CLAS at Jefferson Lab.) The non-anomalous processes n —> 3n is especially interesting because it is sensitive to the isospin violation. While the average u and d-quark mass, (mu + md)/2, is well known, there exists significant uncertainty in their mass difference, md—mu. The n —> 3n decay is particularly suitable for md — mu difference determination because of the suppressed electromagnetic contributions [10,11].
Since the microscopic understanding of strongly interacting matter (both in hadronic phase and in quark-gluon phase) is of great importance also for the physics of heavy ion collisions and compact stars, extending the quark propagator Ansatz with good analytic structure to finite densities and temperatures should also be investigated. This is necessary, for example (to name one concrete task), for extending our analyses of the n-n' complex [12,13] to finite densities and temperatures. Of particular interest is extending to finite density our analysis of the possible UA(1) symmetry restoration [14] in the n-n' complex.
Acknowledgement
This work has been supported in part by the Croatian Science Foundation under the project number 8799. The authors acknowledge the partial support of the COST Action MP1304 Exploring fundamental physics with compact stars (New-CompStar).
References
1. C. D. Roberts and S. M. Schmidt, Prog.Part.Nucl.Phys. 45 (2000) S1-S103.
Analytic structure of nonperturbative quark propagators ... 19
2. R. Alkofer and L. von Smekal, Phys.Rept. 353 (2001) 281.
3. D. Kekez and D. Klabucar, Phys. Rev. D 71 (2005) 014004 [hep-ph/0307110], and our references therein.
4. R. Alkofer, W. Detmold, C. Fischer, and P. Maris,Phys.Rev. D70 (2004) 014014.
5. Y. Jiang et al, Phys.Rev. C78 (2008) 025214; Y. Jiang et al., Phys.Rev. D78 (2008) 116005.
6. U. Habel, R. Konning, H. Reusch, M. Stingl, and S. Wigard, Z.Phys. A336 (1990) 435-447.
7. U. Habel, R. Konning, H. Reusch, M. Stingl, and S. Wigard, Z.Phys. A336 (1990) 423-433.
8. S. Benic, D. Blaschke and M. Buballa, Phys. Rev. D 86 (2012) 074002.
9. R. Oehme and W. t. Xu, Phys. Lett. B 384 (1996) 269.
10. D. Sutherland, Phys.Lett. 23 (1966) 384.
11. J. Bell and D. Sutherland, Nucl.Phys. B4 (1968) 315-325.
12. For analytic, closed form results for the masses and mixing in the t|-t|' complex, and for our earlier results on the r|-T|' complex, see [13].
13. S. Benic, D. Horvatic, D. Kekez and D. Klabucar, Phys. Lett. B 738 (2014) 113.
14. S. Benic, D. Horvatic, D. Kekez and D. Klabucar, Phys. Rev. D 84, 016006 (2011) [arXiv:1105.0356 [hep-ph]], and our references therein.
Bled Workshops in Physics Vol. 16, No. 1 p. 20
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Comparing mesons and WLWL TeV-resonances*
Antonio Dobadoa, Rafael L. Delgadoa, Felipe J. Llanes-Estradaa and Domenec Espriub
a Dept. Fisica Teorica I, Univ. Complutense, 28040 Madrid, Spain b Institut de Ciencies del Cosmos (ICCUB), Marti Franques 1, 08028 Barcelona, Spain
Abstract. Tantalizing LHC hints suggest that resonances of the Electroweak Symmetry Breaking Sector might exist at the TeV scale. We recall a few key meson-meson resonances in the GeV region that could have high-energy analogues which we compare, as well as the corresponding unitarized effective theories describing them. While detailed dynamics may be different, the constraints of unitarity, causality and global-symmetry breaking, incorporated in the Inverse Amplitude Method, allow to carry some intuition over to the largely unmeasured higher energy domain. If the 2 TeV ATLAS excess advances one such new resonance, this could indicate an anomalous q q W coupling.
1 Non-linear EFT for WlWl and hh
The Electroweak Symmetry Breaking Sector of the Standard Model (SM) has a low-energy spectrum composed of the longitudinal W±, ZL and the Higgs-like h bosons. Various dynamical relations suggest that the longitudinal gauge bosons are a triplet under the custodial SU(2)c, and h is a singlet. This is analogous to hadron physics where pions fall in a triplet and the n meson is a singlet. The global symmetry breaking pattern, SU(2) x SU(2) —» SU(2)c is shared between the two fields.
The resulting effective Lagrangian, employing Goldstone bosons w a ~ WL, ZL as per the Equivalence Theorem (valid for energies sufficiently larger than MW, MZ), in the non-linear representation, is [1-3],
L = 2
1 1 h /hx 2 1 +2a—+b -v V v
\ 1
3^3^ j Sij + v"2 ) +
v4 v4 v4
+ 3^H3^h3v^i3v + (1)
This Lagrangian is adequate to explore the energy region 1-3 TeV > 100 GeV, and contains seven parameters. Their status is given in [1] and basically amounts to
Talk delivered by Felipe J. Llanes-Estrada
Comparing mesons and WLWL TeV-resonances
21
a € (0.88,1.3) (1 in the SM), b G (-1,3) (1 in the SM) and the other, NLO parameters (vanishing in the SM) largely unconstrained. This is a reasonably manageable Lagrangian for LHC exploration of electroweak symmetry breaking in the TeV region, before diving into the space of the fully fledged effective theory [3].
Partial wave scattering amplitudes in perturbation theory A{ (s) = A J O' (s) + ajlo' (s)... for ww and hh, have been reported to NLO in [4]. For example, the LO amplitudes of I = 0,1 and 2, and the ww —» hh channel-coupling one are
A0(s) = ^(1 - a2)s A1(s) = 96^(1 - a2)s 1 /3
A0(s) = -32nV2(1 - a2)s M°(s) = ^(a2 - b)s
and we see how any small separation of the parameters from the SM value a2 = b = 1 leads to energy growth, and eventually to strong interactions. To NLO, the amplitudes closely resemble those of chiral perturbation theory
A JO+NLO' (s) = Ks + (V) + D log ^ + E log -) s2 (2)
with a left cut carried by the Ds2 log s term, a right cut in the Es2 log(-s) term, and the Ks + Bs2 tree-level polynomial. B, D and E can be found in [4] and satisfy perturbative renormalizability (in the chiral sense).
2 Resonances
The perturbative amplitudes in Eq. (2) do not make sense for large s (TeV-region) where they violate unitarity ImAj = |Aj |2, relation satisfied only order by order in perturbation theory, namely ImAjLO' = |ajO'|2.
In hadron physics, the solution is to construct new amplitudes that satisfy unitarity exactly and reproduce the effective theory at low energy (see the lectures [5]) via dispersive analysis. This combination of dispersion relations with effective theory exploits all model-independent information in the two-body experimental data, and is known in both the electroweak symmetry breaking sector and the QCD sector of the Standard Model [7]. A salient example is the NLO Inverse Amplitude Method,
A K°')2 /3)
J A(LO' A(NLO' (3)
AIJ - AIJ
a simple formula that can be rigorously generalized to two channels of massless particles by upgrading the various A to matrices. The denominator of Eq. (3) allows for scattering resonances (poles in the 2nd Riemann sheet).
In meson physics, the most salient elastic resonance of the nn system is the isovector p(770) meson, that dominates low-energy dipion production in most experiments; for example, its prominence in COMPASS data [6] is visible in the
22
Felipe J. Llanes-Estrada et al.
w14000
a>
S12000
®
s10000
c
® 8000 6000 4000 2000
0^05
1 1.5 2 2.5 3 3.5 inv. mass of both rc'rc0 system [GeV/c 2|
10.8-¿0.60.40.2-
0,
0
— ' \ ...
a = 0 a- = +0.005
2
s (TeV2)
4 5
Fig. 1. Left: the physical p in the COMPASS nn spectrum. (Reprinted from [6]. Copyright 2008, AIP Publishing LLC). Right: a possible equivalent WW, WZ state for various a4, a5.
a4 = a5 = 0
a, = +0.005 a. =0
4 5
a4 = -0.005 a5 = 0
a4 = 0 a5 = -0.005
2
3
left plot of figure 1. Independently of particular technicolor models, values of a4 and a5 at the 10-4-10-3 level produce a p-like meson of the electroweak sector in the TeV region. The right panel of figure 1 demonstrates this.
The central attraction of the nuclear potential suggested the introduction of a scalar a meson in the nn spectrum whose existence was long disputed but that is now well established [8]. In addition to detailed dispersive analysis, it gives strength to the low-energy nn spectrum if the p channel is filtered out by cautious quantum number choice, such as J/^ —» wnn that forces the pion subsystem to have positive charge conjugation because the other two mesons both have C = (—1). An analysis of BES data by D. Bugg is shown in the left plot of fig. 2. The right plot shows the equivalent resonance in ww ~ WLWL, that appears for
1=-0.80.60.40.2-
■ IAM
----N/D
------K-improved
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
E (TeV)
Fig. 2. Left: nn spectrum with positive charge conjugation clearly showing an enhancement at low invariant mass, related to the fo(500) (or a) meson; (Reprinted from [9] with permission. Copyright 2008, AIP Publishing LLC). Right: IJ = 00 ww scattering in the IAM and other unitarization methods producing an equivalent electroweak resonance.
Comparing mesons and WLWL TeV-resonances
23
a = 1 and/or b = a2 (if the resonance is induced by b alone it is a pure coupled channel one [4], that also has analogues in hadron physics, though less straightforward ones).
The same BES data also reveals another salient meson resonance, the f2 (1270). Partial waves with J = 2 cannot be treated with the NLO IAM, as A0 LO = 0 but a similar structure has been obtained with the N/D or K-matrix unitarization methods, and we show it in figure 3.
0.8-L
0.6-
0.4-
0.2-
N/D ■ K-improved
M i i ¡i i i i i i i i
i ¡ i i i '
i i/ I \l
¡ A.
0
0 0.5 1
1.5 2 2.5 E (TeV)
3 3.5
Fig. 3. Left: generating an IJ = 02 resonance in the electroweak sector is possible with adequate values of a4, a5. Right: positive values of a2 — 1 also generate an isotensor I = 2 resonance, though this is more disputed [2]. In hadron physics the isotensor wave is repulsive, and thus, not resonant.
1
4
3 ATLAS excess in two-jet events
Renewed interest in TeV-scale resonances is due to a possible excess in ATLAS data [10] plotted in figure 4 together with comparable, older CMS data [11] that does not show such an enhancement. The excess is seen in two-jet events tagged as vector boson pairs by invariant mass reconstruction (82 and 91 GeV respectively). The experimental error makes the identification loose, so that the three-channels cross-feed and we should not take seriously the excess to be seen in all three yet. Because WZ is a charged channel, an I = 0 resonance cannot decay there. Likewise ZZ cannot come from an I = 1 resonance because the corresponding Clebsch-Gordan coefficient (1010|10) vanishes. A combination of both I = 0,1 could explain all three signals simultaneously (as would also an isotensor I = 2 resonance).
A relevant relation imported from hadron physics that the IAM naturally incorporates restricts the width of a vector boson. This one-channel KSFR relation [12] links the mass and width of the vector resonance with the low-energy constants v and a in a quite striking manner,
rIAM = MnV2(1 - a2) . (4)
24
Felipe J. Llanes-Estrada et al.
For M ~ 2 TeV and r ~ 0.2 TeV (see fig. 4), we get a ~ 0.73 which is in tension with the ATLAS-deduced bound a^ > 0.88 at 4-5ct level; Eq. (4) predicts that a 2 TeV J = 1 resonance, with current low-energy constants, needs to have r < 50 GeV, a fact confirmed by more detailed calculations [1,2]. However scalar resonances tend to be substantially broader.
1000
o
100
^ 10 rn 1 0.1
: • ! •
■ ■
" ! i
1
1.5
Ejj (TeV)
2.5
Fig. 4. Left: replot of the ATLAS data [10] for WZ -> 2 jet, with a slight excess at 2 TeV (also visible in the other isospin combinations WW and ZZ, not shown). The jet analysis is under intense scrutiny [15]. Right: equivalent CMS data [11] with vector-boson originating jets. No excess is visible at 2 TeV (though perhaps some near 1.8-1.9 TeV).
10000
2
3
The cross section for the reaction pp —» W+Z+X for a given WZ Mandelstam s, and with the E2 total energy in the proton-proton cm frame, can be written [13] in standard LO QCD factorization as
dads
dxu
dxa5(s - xuxaEt20tf(xu)f(xa)a(ud^ w+z) .
(5)
The parton-level cross section a is calculated, with the help of the factorization theorem, from the effective Lagrangian in Eq. (1) above. Following [13], we would
Fig. 5. Production of a pair of Goldstone bosons by ud annihilation through a W-meson and anomalous BSM vertex enhancing it.
expect an amplitude (from the left diagram of figure 5) given by M = uytM-ig^v/8)2(i/q2)(k1 — in perturbation theory. Further, dispersive analysis reveals the need of a vector form factor in the presence of strong final state rescattering, to guarantee Watson's final state theorem; the phase of the production amplitude must be equal to that of the elastic ww scattering amplitude. If the later is represented by the Inverse Amplitude Method, the form factor in the
Www vertex is FV(s) =
A
(i )
(s)
. (0)r
The resulting cross section [13] was
found to be slightly below the CMS bound, and perhaps insufficient to explain
-1
0
0
+
1
Comparing mesons and WLWL TeV-resonances
25
the possible ATLAS excess. With current precision this statement should not be taken to earnestly, but it is nonetheless not too soon to ask ourselves what would happen in the presence of additional non-SM fermion couplings.
Thus, an original contribution of this note is to add to Eq. (1) a term 1
Lfermion anomalous = 'Lw (6)
(for a derivation see, e.g. [14]). The parton level cross-section is then dt"
an
1 g4 sin2 0(l + ^S) Vv(s)|2 ,
cm
64n2 s 32
(7)
and if Si = 0 additional production strength appears in the TeV region. The sign of this Si might be determined from the line shape due to interference with the background [16].
4 Conclusion
110
1.5 2 E (TeV)
110
110
110"
110
110-5
310
Fig. 6. Tree-level W production of œœ [13] with final-state resonance; non-zero parameters are a=0.9, b= a2, a4 = 7x 10-4 (at h = 3 TeV). Also shown is a CMS cross-section upper bound (see fig. 4). This can be exceeded with the 5i coupling of Eq. (6).
The 13 TeV LHC run II entails larger cross sections and allows addressing the typical tr, p-like ww resonances, at the edge of the run I sensitivity limit as shown in fig. 6. The large rate at which such a resonance would have to be produced to explain the ATLAS excess (at the 10fbarn level [17]) is a bit puzzling, though it can be incorporated theoretically with the Si parameter. Hopefully this ATLAS excess will soon be refuted or confirmed. In any case, the combination of effective theory and unitarity that the IAM encodes is a powerful tool to describe data up to E = 3 TeV in the electroweak sector if new, strongly interacting phenomena
1 This is only one of the possible additional operators. There is a second one with R fields, and several custodially breaking others. The gauge-invariant version of Eq. (6) actually modifies the fermion-gauge coupling by a factor (1+S1): this cannot be excluded because it would be the quantity that is actually well measured in | decay. The triple gauge
boson vertex would then need not coincide with this coupling. However the latter is much less precisely known and there is room for deviations at the 5-10% level.
26 Felipe J. Llanes-Estrada et al.
appear, with only few independent parameters. The content of new, Beyond the Standard Model theories, can then be matched onto those parameters for quick tests of their phenomenological viability.
Acknowledgements
FLE thanks the organizers of the Bled workshop "Exploring hadron resonances" for hospitality and encouragement. Work supported by Spanish Excellence Network on Hadronic Physics FIS2014-57026-REDT, and grants UCM:910309, MINEC0:FPA2014-53375-C2-1-P, FPA2013-46570,2014-SGR-104, MDM-2014-0369; its completion was possible at the Institute for Nuclear Theory of the Univ. of Washington, Seattle, with D0E support.
References
1. R.L.Delgado, A.Dobado and F.J.Llanes-Estrada, J. Phys. G 41, 025002 (2014); ibid, JHEP 1402,121 (2014); R.L.Delgado, A.Dobado, M.J.Herrero and J.J.Sanz-Cillero, JHEP 1407, 149 (2014).
2. P. Arnan, D. Espriu and F. Mescia, arXiv:1508.00174 [hep-ph]. D. Espriu and F. Mescia, Phys. Rev. D 90, 015035 (2014). D. Espriu, F. Mescia and B. Yencho, Phys. Rev. D 88, 055002 (2013). D. Espriu and B. Yencho, Phys. Rev. D 87, no. 5, 055017 (2013).
3. R. Alonso et al, JHEP 1412, 034 (2014); G.Buchalla, O.Cata, A.Celis and C.Krause, arXiv:1504.01707 [hep-ph].
4. R. L. Delgado, A. Dobado and F. J. Llanes-Estrada, Phys. Rev. Lett. 114, 221803 (2015).
5. T. N. Truong, EFI-90-26-CHICAG0, EP-CPT-A965-0490, UCSBTH-90-29, C90-01-25.
6. F. Nerling [COMPASS Collaboration], AIP Conf. Proc. 1257, 286 (2010) [arXiv:1007.2951 [hep-ex]].
7. A. Dobado, M. J. Herrero and T. N. Truong, Phys. Lett. B 235,129 (1990); A. Dobado and J. R. Pelaez, Nucl. Phys. B 425, 110 (1994) [Nucl. Phys. B 434, 475 (1995)] [hep-ph/9401202].
8. J. R. Pelaez, Phys. Rept.(in press) arXiv:1510.00653 [hep-ph].
9. D. V. Bugg, AIP Conf.Proc.1030,3 (2008) [arXiv:0804.3450 hep-ph].
10. G. Aad etal. [ATLAS Collaboration], arXiv:1506.00962 [hep-ex].
11. V. Khachatryan et al. [CMS Collaboration], JHEP 1408,173 (2014).
12. R. L. Delgado, A. Dobado and F. J. Llanes-Estrada, Phys. Rev. D 91, 075017 (2015).
13. A. Dobado, F. K. Guo and F. J. Llanes-Estrada, Commun. Theor. Phys. (in press), arXiv:1508.03544 [hep-ph].
14. E. Bagan, D. Espriu and J. Manzano, Phys. Rev. D 60,114035 (1999).
15. D. Goncalves, F. Krauss, M. Spannowsky, arXiv:1508.04162 [hep-ph].
16. C. H. Chen and T. Nomura, arXiv:1509.02039 [hep-ph].
17. ATLAS contribution to the 3rd Annual LHC Physics Conference, St. Petersburg, 31/8 to 5/9 2015, ATLAS-C0NF-2015-045.
Bled Workshops in Physics Vol. 16, No. 1 p. 27
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Resonances in the Constituent-Quark Model*
R. Kleinhappel and W. Plessas
Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria
Abstract. We give a short account of the present description of baryon resonances within the relativistic constituent-quark model, where resonances are usually treated as excited bound states, and point to ways for a more realistic theory producing the resonances as complex poles in the momentum/energy planes, i.e. with real mass values and finite widths.
Nowadays the relativistic constituent-quark model, especially the one relying on a dynamics of linear confinement and a hyperfine interaction of Goldstone-boson exchange [1,2], can provide a reasonable description of the baryon spectra (see, e.g., the recent review in ref. [3]. It is even possible to reproduce - with only a few exceptions - the real mass values of all known baryon ground and resonant states with flavors u, d, s, c, and b in a universal framework in close agreement with phenomenology or data from lattice chromodynamics (QCD) [4-7]. Herein baryons are considered as relativistic bound states of three confined constituent quarks Q interacting mutually. The Q's are supposed to be quasi-particles with dynamical masses generated by the spontaneous breaking of chiral symmetry (SBxS) of low-energy QCD [8,9]. The most important ingredients in the three-Q invariant mass operator turn out to be SBxS and relativistic invariance [10].
Solving the three-Q mass-operator eigenvalue problem provides also access to the baryon eigenfunctions (see, e.g., their rest-frame spatial representations in ref. [12]). They can be subject to tests in various baryon reactions. While their structures appear to be quite reasonable for the baryon ground states, the resonance wave functions are obviously affected by shortcomings.
In particular, the electromagnetic form factors of the nucleons as well as their electric radii and magnetic moments are reproduced in good agreement with phenomenology [13], even with regard to their flavor contents [14]. Similarly, the electromagnetic form factors, electric radii, and magnetic moments of the A and hyperon ground states are found well compatible with available data from experiment and lattice QCD [15,16]. The same is true with regard to the axial form factors and axial charges [16-18]. Likewise, the gravitational form factor A(Q2) of the nucleons is reasonably reproduced [3]. In addition, for the strong form factors of the nNN and nNA a microscopic explanation is provided that conforms with the ones usually adopted in nN and nA dynamical models [19].
* Talk delivered by by W. Plessas
28
R. Kleinhappel and W. Plessas
Disturbing shortcomings of the three-Q constituent-quark model appear with regard to direct predictions for hadronic decays of the n, n, and K meson modes. First fully relativistic results in general show an undershooting of the experimental decay widths [12,20-22]. This hints to missing ingredients in the adopted approach. The problems may either be connected with an improper treatment of the meson-decay vertex or missing degrees of freedom from the decay channels.
In order to remedy the situation we have recently adhered to a coupledchannels formalism, taking into account mesonic decays channels explicitly. First attempts along a toy model for mesons yielded promising results [23]. At least it could be shown, how finite resonance widths well develop in such an approach.
Further studies concerned explicit pionic effects both on the ground state and resonance masses (including the resonance widths), exemplified for the nucleon and the A. Details of the formalism and first results were presented at previous Bled Workshops [24-26]. While the pionic effects on the nucleon mass appear reasonable, the analogous treatment of the A does not yet enhance especially its n-decay width enough in order to make it compatible with the relatively large phenomenological value [26]. Further ingredients appear to be necessary. We are presently in the course of extending the coupled-channels theory accordingly.
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. L. Y. Glozman, W. Plessas, K. Varga and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998)
2. L. Y. Glozman, Z. Papp, W. Plessas, K. Varga and R. F. Wagenbrunn, Phys. Rev. C 57, 3406 (1998)
3. W. Plessas, Int. J. Mod. Phys. A 30, no. 02,1530013 (2015)
4. J. P. Day, W. Plessas, and K. S. Choi, arXiv:1205.6918 [hep-ph]
5. J. P. Day, K. S. Choi, and W. Plessas, Few-Body Syst. 54, 329 (2013)
6. J. P. Day, W. Plessas, and K. S. Choi, in: Looking into Hadrons (Proceedings of the MiniWorkshop, Bled, Slovenia, 2013), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2013); p. 6
7. J. P. Day, PhD Thesis, University of Graz (2013)
8. W. Plessas, in: Quark Masses and Hadron Spectra (Proceedings of the Mini-Workshop, Bled, Slovenia, 2014), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2014); p. 34
Resonances in the Constituent-Quark Model
29
9. M. Rosina, in: Quark Masses and Hadron Spectra (Proceedings of the Mini-Workshop, Bled, Slovenia, 2014), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2014); p. 50
10. W. Plessas, Mod. Phys. Lett. A 28, no. 26,1360022 (2013)
11. W. Plessas, PoS LC 2010, 017 (2010); arXiv:1011.0156 [hep-ph]
12. T. Melde, W. Plessas and B. Sengl, Phys. Rev. D 77,114002 (2008)
13. R. F. Wagenbrunn, S. Boffi, W. Klink, W. Plessas, and M. Radici, Phys. Lett. B 511, 33 (2001)
14. M. Rohrmoser, K. S. Choi, and W. Plessas, Acta Phys. Polon. Supp. 6, 371 (2013)
15. K. Berger, R. F. Wagenbrunn, and W. Plessas, Phys. Rev. D 70, 094027 (2004)
16. K. S. Choi and W. Plessas, Few-Body Syst. 54,1055 (2013)
17. L. Y. Glozman, M. Radici, R. F. Wagenbrunn, S. Boffi, W. Klink, and W. Plessas, Phys. Lett. B 516,183 (2001)
18. S. Boffi, L. Y. Glozman, W. Klink, W. Plessas, M. Radici, and R. F. Wagenbrunn, Eur. Phys. J. A 14,17 (2002)
19. T. Melde, L. Canton, and W. Plessas, Phys. Rev. Lett. 102,132002 (2009)
20. T. Melde, W. Plessas, and R. F. Wagenbrunn, Phys. Rev. C 72, 015207 (2005); ibid. C 74, 069901 (2006)
21. B. Sengl, T. Melde, and W. Plessas, Phys. Rev. D 76, 054008 (2007)
22. T. Melde, W. Plessas, and B. Sengl, Phys. Rev. C 76, 025204 (2007)
23. R. Kleinhappel, W. Plessas, and W. Schweiger, Few-Body Syst. 54, 339 (2013)
24. R. Kleinhappel, W. Plessas, and W. Schweiger, in: Understanding Hadron Spectra (Proceedings of the Mini-Workshop, Bled, Slovenia, 2011), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2011); p. 36
25. R. Kleinhappel, W. Plessas, and W. Schweiger, in: Hadronic Resonances (Proceedings of the Mini-Workshop, Bled, Slovenia, 2012), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2012); p. 20
26. R. Kleinhappel, L. Canton, W. Plessas, and W. Schweiger, in: Quark Masses and Hadron Spectra (Proceedings of the Mini-Workshop, Bled, Slovenia, 2014), ed. by B. Golli, M. Rosina, and S. Sirca. DMFA, Ljubljana (2014); p. 22
Bled Workshops in Physics Vol. 16, No. 1 p. 30
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Unquenched quark-model calculation of excited p resonances and P-wave nn phase shifts*
Susana Coitoa, George Ruppb, Eef van Beverenc "Institute of Modern Physics, CAS, Lanzhou 730000, China
bCeFEMA, Instituto Superior Tecnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal cCentro de Física Computational, Departamento de Física, Universidade de Coimbra, 3004-516 Coimbra, Portugal
Abstract. The p(770) vector resonance, its radial recurrences, and the corresponding P-wave nn phase shifts are investigated in an unquenched quark model with all classes of relevant decay channels included, viz. pseudoscalar-pseudoscalar, vector-pseudoscalar, vector-vector, vector-scalar, axialvector-pseudoscalar, and axialvector-vector, totalling 26 channels. Two of the few model parameters are fixed at previously used values, whereas the other three are adjusted to the p(770) resonance and the lower P-wave nn phases. Preliminary results indicate the model's capacity to reproduce these phases as well as the p mass and width. However, at higher energies the phase shifts tend to rise too sharply. A possible remedy is an extension of the model so as to handle resonances in the final states for most of the included decay channels. Work in progress.
1 Introduction
The radial recurrences of the p(770) vector resonance play a crucial role in lightmeson spectroscopy, owing to the several observed states in the PDG tables [1], up to 1.9 GeV, and the corresponding P-wave pion-pion phase shifts and inelasticities measured in several experiments [1]. These resonances may shed a lot of light on the underlying quark-confinement force as well as the strong-decay mecanism, both assumed to result from low-energy QCD. However, there are two serious problems, one experimental and the other theoretical. First of all, excited p states listed in the PDG tables are far from well established, even the generally undisputed p(1450) [1] resonance. For example, under the entry of the latter state in the PDG meson listings, experimental observations have been collected with masses in the range 1250-1582 MeV. The state of lowest mass here corresponds to the resonance at 1.25 GeV first observed by Aston et al. in 1980 [2]. Several other experiments have confirmed such a vector p' resonance in the mass interval 1.25-1.3 GeV, both in the [3] and nn [4] channels. More recently, a combined S-matrix and Breit-Wigner (BW) analysis [5], using P-wave nn data from the early 1970s, not only confirmed a p(1250), but even found it to be much
* Talk delivered by George Rupp
Unquenched quark-model calculation of excited p resonances
31
more important for a good fit to the data than the p(1450). In view of these findings, it is simply inconceivable that no separate p(1250) entry has been created in the PDG tables, and to make things worse, the latter analysis [5] is not even included in the PDG references. The reason for these omissions seems to be based on theory bias, which is the second problem. Indeed, the renowned relativised quark model for mesons by Godfrey & Isgur [6] predicted the first radial p excitation at 1.45 GeV. As a matter of fact, practically all quark models employing the usual Coulomb-plus-linear confining potential find a p' at about the same mass and cannot accommodate a p(1250) (see, however, Ref. [7]).
Unfortunately, no fully unquenched lattice calculations including both qq and two-meson interpolating fields have been carried out so far beyond the ground-state p(770) [8]. Nevertheless, in the strange-meson sector such a calculation was done recently [9], reproducing the K*(892) resonance in P-wave elastic Kn scattering, with mass and width close to the experimental values. Moreover, the first radial excitation was identified as well, tentatively at 1.33 GeV, which should correspond to the K* (1410) [1] resonance. The latter is thus determined on the lattice to be a normal quark-antiquark resonance. From simple quark-mass considerations, one is led to conclude the same for the p' originally found by Aston et al. [2] at 1.25 GeV, which contradicts the speculation in Ref. [10] that it "has necessarily to be an exotic". Note that including meson-meson interpolators is absolutely crucial to reliably predict the mass of an excited resonance like the K*', since an also unquenched lattice calculation without considering decay, by partly the same authors [11], predicted a K*' mass a full 300 MeV heavier than in Ref. [9].
In the present work, we analyse the issue of radial p recurrences by attempting to describe the elastic and inelastic P-wave nn phase shifts in the context of an unquenched quark model that has been successfully applied to several problematic mesons (see e.g. Ref. [12] for a very brief recent review). In this so-called Resonance-Spectrum-Expansion (RSE) model, the manifest non-perturbative inclusion of all relevant two-meson channels alongside a confined qq sector allows for a phenomenlogical decription of excited meson resonances in the same spirit as the referred lattice calculation [9]. An additional advantage is that the RSE model yields an exactly unitary and analytic S-matrix for any number of included quark-antiquark and two-meson channels, whereas the lattice still faces serious problems in the case of inelastic resonances and highly excited states.
2 RSE modelling of P-wave nn scattering
The general expressions for the RSE off-energy-shell T-matrix and corresponding on-shell S-matrix have been given in several papers (see e.g. Ref. [13]). In the present case of P-wave nn scattering, the quantum numbers of the system are IG JPC = 1+ 1 , which couples to the I = 1 quark-antiquark state (uu-dd)/%/2 in the spectroscopic channels 3Si and 3Di. In the meson-meson sector, we only consider channels allowed by total angular momentum J, isospin I, parity P, and G-parity G. The included combinations are pseudoscalar-pseudoscalar (PP), vector-pseudoscalar (VP), vector-vector (VV), vector-scalar (VS), axialvector-pseudoscalar (AP), and axialvector-vector (AV), with mesons from the lowest-lying pseudoscalar,
32
Susana Coito, George Rupp, Eef van Beveren
vector, scalar, and axialvector nonets listed in the PDG [1] tables. Here, "axialvec-tor" may refer to JPC = 1++, JPC = 1 + , or JP = 1 + for mesons with no definite C-parity. This choice of meson-meson channels is motivated by the observed two-and multi-particle decays of the p recurrences up to the p(1900) [1], which include several intermediate states containing resonances from the referred nonets. For instance, the PDG lists [1] under the 4n decays of the p(1450) the modes wn, a (1260)n, hi (1170)n, n(1300)n, pp, and p(nn)s_wave, where (nn)s_wave is probably dominated by the fo(500) [1] scalar resonance. By the same token, the 6n decays of the p(1900) will most likely include important contributions from modes as bi (1235) p, ai (1260) .... For consistency of our calculation, we generally include complete nonets in the allowed decays, and not just individual modes observed in experiment. The only exception is the important n(1300)n P'P mode, because no complete nonet of radially excited pseudoscalar mesons has been observed so far [1]. The included 26 meson-meson channels are given in Table 1.
Nonets Two-Meson Channels L
PP nn, KK 1
VP wn, pr|, pn', K*K 1
VV pp, K*K* 1
VS pfo(500), wao(980), K*KJ(800) 0,2
AP ai (1260)n, bi (1235)n, bi (1235)n', hi (1170)n, Ki (1270)K, Ki (1400)K 0
AV ai (1260) w, bi (1235)p, fi (1285)p, Ki (1270)K*, Ki (1400)K* 0
P'P n(1300)n 1
Table 1. Included classes of decay channels containing mesons listed [1] in the PDG tables, with the respective orbital angular momenta. Note that the included P'P decay mode is incomplete (see text above).
Notice that the VS channels count twice, because they can have L = 0 or L = 2. However, the S-wave only couples to the 3S1 qq channel and the D-wave only to the 3D1. The relative couplings between the qq and meson-meson channels are determined using the scheme of Ref. [14], based on overlaps of harmonic-oscillator (HO) wave functions. Special care is due in the cases of flavour mixing (n,V), and mixing of the C-parity eigenstates 3P1 and iP1 [13], as the strange axialvector mesons K1 (1270) and K1 (1400) have no definite C-parity.
Coming now to describing the data, we have to adjust the model parameters. Two of these, namely the non-strange constituent quark mass and the HO oscillator frequency, are as always fixed [13] at the values mn = mu = md = 406 MeV and w = 190 MeV. This yields a largely degenerate bare p spectrum with energy levels 1097 MeV (1 3Si), 1477 MeV (23Si/1 3Di), 1857 MeV (3 3Si/2 3Di), 2237 MeV (43S1/33D1), ... . This bare spectrum is then deformed upon un-quenching, that is, by allowing q q pair creation. This results in real or complex mass shifts due to meson-loop contributions, depending on decay channels being closed or open, respectively. Note that these shifts are non-perturbative and can
Unquenched quark-model calculation of excited p resonances
33
only be determined, for realistic coupling strengths, by numerically finding the poles of the S-matrix. The adjustable parameters are the overall dimensionless coupling constant A, the "string-breaking" radius r0 for transitions between the q q and two-meson channels, and a range parameter a for weakening subthreshold contributions via a form factor. The coupling A is usually in the range 3-5, r0 should be of the order of 1 fm for systems made of light quarks (with mass mn), and the a value used in several previous papers is 4 GeV-2 (see e.g. Ref. [15]).
In view of these strong limitations, it is quite remarkable that we can obtain a good fit to the P-wave nn phase shifts up to 1.2 GeV with the choice A = 5.3, r0 = 0.9 fm, and a = 4 GeV-2. Moreover, the corresponding p(770) pole comes out at the very reasonable energy E = (754-i67) MeV. At higher energies, though, the nn phases tend to rise too fast and no good description has been obtained so far, also due to the very little fitting freedom. As for the poles of the higher p recurrences, we find at least four in the energy range 1.2-2.0 GeV, in agreement with the PDG [1] and also Ref. [5], albeit at yet quite different energies. In particular, there are two poles between 1.2 and 1.5 GeV, in qualitative agreement with Ref. [5]. However, these pole positions are extremely sensitive to the precise values of the parameteres A, r0, and a, so that they should not be taken at face value as long as no good fit is achieved of the observables above 1.2 GeV.
A possibility to improve the fit is by allowing different decay radii for the several classes of decay channels (PP, VP, VV, VS, AP, AV, P'P), which would be logical in view of the detailed, channel-dependent transition potentials derived in Ref. [14]. Such an additional flexibility will neither affect the exact solvability of the RSE T-matrix, nor its analyticity.
An addditional possible model extension we shall discuss in the next section.
3 Resonances in asymptotic states
The decay channels listed in Table 1 contain several resonances, some of which are even extremely broad, with widths exceeding 300 MeV. So treating the corresponding thresholds as being sharp, at well-defined real energies, is certainly an approximation, which may produce too sudden effects at threshold openings. Ideally, one would like to describe a resonance in the final state via a smooth function of real energy, corresponding to an experimental cross section in which the resonance is observed. By discretising such a function, one could in principle describe each final-state resonance through a large number of effective thresholds. However, this would lead to a proliferation of channels and to a true explosion of Riemann sheets, making the tracing of complex poles impracticable.
An alternative way to handle a resonance in asymptotic states is to replace its real mass by a complex one, on the basis of the PDG [1] resonance mass and total width. However, this inevitably destroys unitarity of the S-matrix. Nevertheless, its symmetry will be unaffected, which can be used to define a new matrix that is unitary again, and so take over the role of S. Here, we closely follow the derivation given in Ref. [16].
34
Susana Coito, George Rupp, Eef van Beveren
An arbitrary symmetric matrix S can be decomposed, via Takagi [17] factorisation, as
S = VDVt , (1)
where V is unitary and D is a real non-negative diagonal matrix. Then we get
S tS = (VT )tDVtVDVT = (VT )fD2VT = UfD2U, (2)
where we have defined U = VT, which is obviously unitary, too. So the diagonal elements of D = VuSñSUt are the square roots of the eigenvalues of the positive Hermitian matrix St S, which are all real and non-negative. Moreover, since S =
1 + 2iT is manifestly non-singular, the eigenvalues of StS are even all non-zero and U is unique. Thus, we may define
S' = S UtD-1U. (3)
Then, using Eq. (1) and V = UT, we have
S' = UT DUUtD-1U = UT U, (4)
which is obviously symmetric. But it is also unitary, as
(UT U)t = Ut(Ut )T = U-1 (U-1 )T = (UT U)-1 . (5)
So S' has the required properties to be defined as the scattering matrix for a process with complex masses in the asymptotic states. Note that this empirical method has been applied very successfully to the enigmatic X(3872) charmonium state in Ref. [16].
4 Conclusions
We have presented preliminary results of an unquenched quark-model study aimed at determining the complex pole positions of vector p recurrences up to
2 GeV, motivated by the poor status of these resonances in the PDG tables [1] and their importance for light-meson spectroscopy. The employed RSE model was applied in the past to a variety of problematic mesons, with very good results [12]. The here included classes of meson-meson channels cover most of the observed strong decays. The three adjustable model parameters were fitted to the P-wave nn phase shifts up to about 1.2 GeV, allowing a good reproduction of these data and a very reasonable p(770) resonance pole position.
However, at energies above 1.2 GeV the thus calculated phases rise too fast, and a globally good fit including the higher nn phases is not feasible with only three parameters. A possible model extension amounts to allowing different decay radii for the different classes of meson-meson channels, which will not spoil the nice model features. Another extension to be considered is the use of complex physical masses for the final-state resonances in nearly all channels of Table 1. This will require a redefinition of the S-matrix so as to restore manifest unitarity, which can be done with an empirical algebraic procedure, exploiting the symmetry of S.
All this work is in progress.
Unquenched quark-model calculation of excited p resonances
35
References
1. K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38 (2014) 090001.
2. D. Aston et al. [Bonn-CERN-Ecole Poly-Glasgow-Lancaster-Manchester-Orsay-Paris-Rutherford-Sheffield Collaboration], Phys. Lett. B 92 (1980) 211 [Erratum ibid. 95 (1980) 461].
3. D. P. Barber et al. [LAMP2 Group Collaboration], Z. Phys. C 4 (1980) 169.
4. L. M. Kurdadze et al, JETP Lett. 37 (1983) 733 [Pisma Zh. Eksp. Teor. Fiz. 37 (1983) 613]; S. Dubnicka and L. Martinovic, J. Phys. G 15 (1989) 1349; D. Aston et al., Nucl. Phys. Proc. Suppl. 21 (1991) 105; A. Bertin et al. [OBELIX Collaboration], Phys. Lett. B 414 (1997) 220.
5. Y. S. Surovtsev and P. Bydzovsky, Nucl. Phys. A 807 (2008) 145.
6. S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189.
7. E. van Beveren, G. Rupp, T. A. Rijken, and C. Dullemond, Phys. Rev. D 27 (1983) 1527.
8. C. B. Lang, D. Mohler, S. Prelovsek, and M. Vidmar, Phys. Rev. D 84 (2011) 054503 [Erratum ibid. 89 (2014) 059903] [arXiv:1105.5636 [hep-lat]].
9. S. Prelovsek, L. Leskovec, C. B. Lang, and D. Mohler, Phys. Rev. D 88 (2013) 5, 054508 [arXiv:1307.0736 [hep-lat]].
10. A. Donnachie, Y. S. Kalashnikova, and A. B. Clegg, Z. Phys. C 60 (1993) 187.
11. G. P. Engel, C. B. Lang, D. Mohler, and A. Schafer, PoS Hadron 2013 (2013) 118 [arXiv:1311.6579 [hep-ph]].
12. G. Rupp, E. van Beveren, and S. Coito, Acta Phys. Polon. Supp. 8 (2015) 139 [arXiv:1502.05250 [hep-ph]].
13. S. Coito, G. Rupp, and E. van Beveren, Phys. Rev. D 84 (2011) 094020 [arXiv:1106.2760 [hep-ph]].
14. E. van Beveren, Z. Phys. C 21 (1984) 291 [hep-ph/0602246].
15. S. Coito, G. Rupp, and E. van Beveren, Phys. Rev. D 80 (2009) 094011 [arXiv:0909.0051 [hep-ph]].
16. S. Coito, G. Rupp, and E. van Beveren, Eur. Phys. J. C 71 (2011) 1762 [arXiv:1008.5100 [hep-ph]].
17. T. Takagi, Japan J. Math. 1 (1924) 82.
Bled Workshops in Physics Vol. 16, No. 1 p. 36
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
The pion-cloud contribution to the electromagnetic nucleon structure*
D. Kupelwieser and W. Schweiger
Institute of Physics, University of Graz, A-8010 Graz, Austria
Abstract. The present contribution continues and extends foregoing work on the calculation of electroweak form factors of hadrons using the point-form of relativistic quantum mechanics. Here we are particularly interested in studying pionic effects on the electromagnetic structure of the nucleon. To this aim we employ a hybrid constituent-quark model that comprises, in addition to the 3q valence component, also a 3q+n non-valence component. With a simple wave function for the 3q component we get reasonable results for the nucleon form factors. In accordance with other authors we find that the pionic effect is significant only below Q2 < 0.5 GeV2.
In a series of papers [1-4] we have developed and advocated a method for the calculation of the electroweak structure of few-body bound states that is based on the point form of relativistic quantum mechanics. All types of interactions are introduced in a Poincare-invariant way via the Bakamjian-Thomas construction [5]. Our strategy is then to determine the invariant 1 -y-exchange (1 -W-exchange) amplitude, extract the electroweak current of the bound state, analyze its covariant structure and determine the form factors. The dynamics of the exchanged gauge boson is thereby fully taken into account by means of a coupled-channel formulation.
Here we are interested in the electromagnetic structure of the nucleon as resulting from a hybrid constituent-quark model in which the nucleon is not just a 3q bound state, but contains, in addition, a 3q+n non-valence component. Transitions between these two components can happen via emission and absorption of the pions by the quarks. In addition, quarks are subject to an instantanous confining force. Typical for the point-form, all four components of the momentum operator are interaction dependent, whereas the generators of Lorentz transformations stay free of interactions. This entails simple rotation and boost properties and angular-momentum addition works like in non-relativistic quantum mechanics. The point-form version of the Bakamjian-Thomas construction allows to separate the overall motion of the system from the internal motion in a neat way:
^ = M fee = (M free + M int) fe , (1)
i.e. the 4-momentum operator factorizes into an interaction-dependent mass operator MM and a free 4-velocity operator Vfee. Bakamjian-Thomas-type mass
* Talk delivered by W. Schweiger
The pion-cloud contribution to the electromagnetic nucleón structure
37
operators are most conveniently represented in terms of velocity states |V; ki, m; k2, n-2',...; kn, ^n), which specify the system by its overall velocity V (V^V^ = 1), the CM momenta kt of the individual particles and their (canonical) spin projections ^t [2].
We now want to calculate the ly-exchange amplitude for elastic electron scattering off a nucleon that consists of a 3q and a 3q+n component. A multichannel formulation that takes not only the dynamics of electron and quarks, but also the dynamics of the photon and the pion fully into account has to comprise all states which can occur during the scattering process (i.e. |3q, e), |3q, n, e), |3q, e,y), |3q, n, e, y)). What one then needs, in principle, are scattering solutions of the mass-eigenvalue equation
(2)
which evolve from an asymptotic electron-nucleon in-state |eN) with invariant mass a/s. The diagonal entries of this matrix mass operator contain, in addition to the relativistic kinetic energies of the particles in the particular channel, an instantaneous confinement potential between the quarks. The off-diagonal entries are vertex operators which describe the transition between the channels. In the velocity-state representation these vertex operators are directly related to usual quantum-field theoretical interaction-Lagrangean densities [2]. Since we only deal with pseudoscalar pion-quark coupling in the following, we have neglected the nyqq-vertex (that would show up for pseudovector pion-quark coupling).
To proceed, we reduce Eq. (2) to an eigenvalue problem for |^3qe) by means of a Feshbach reduction,
/m cone k n K n K Y 0
¡Mconf 3qne 0 0 K Y
K Y Mon y Kn K n
V 0 K Y mm con^eY/
/I^Cqe) \ /I^3qe) \
|^3qne) - ^/s |^3qne)
|^3qeY) — V s |^3qeY)
\I^3qneY)/ \|^3qneY )/
m 3one+K „(vs-
MC3oqnne)-1Kn + Vo^VS)! |^3qe) = v^aqne) , (3)
where VY^Vs] is the ly-exchange optical potential. The invariant ly-exchange electron-nucleon scattering amplitude is now obtained by sandwiching V^iVS between (the appropriately normalized valence component of the) physical electron-nucleon states |eN), i.e. eigenstates of [M?^ + K(Vs - M^f e)-1 Kt]. The crucial point is now to observe that, due to instantaneous confinement, propagating intermediate states do not contain free quarks, they rather contain either physical nucleons N or bare baryons B, the latter being eigenstates of the pure confinement problem. As a consequence one can reformulate the scattering amplitude in terms of pure hadronic degrees of freedom with the quark substructure being hidden in vertex form factors. This is graphically represented in Fig. 1. The analytical expressions for the bare nucleon form factors follow from this reformu-lation.l
1 For details of their extraction and a discussion of the problems connected with wrong cluster properties associated with the Bakamjian-Thomas construction we refer to Refs. [1,2,4].
38
D. Kupelwieser and W. Schweiger
Fig. 1. Diagrams representing the 1y-exchange amplitude for electron scattering off a "physical" nucleon N, i.e. a bare nucleon dressed by a pion cloud. The time orderings of the y-exchange are subsumed under a covariant photon propagator. Black blobs represent vertex form factors for the coupling of a photon or pion to the bare nucleon N. A vertex form factor is also assumed at the photon-pion vertex. The ovals represent the wave function (i.e. essentially the square root of the probability P^ /N) for finding the bare nucleon in the physical nucleon.
If nucleonic excitations are neglected in the pion loop, we just need the electromagnetic yN N and strong nN N vertex form factors (for the bare nucleon N) as well as the electromagnetic pion form factor. The electromagnetic pion form factor can be taken from Ref. [1], where it has been calculated within the same approach as here using a harmonic-oscillator model for the ud bound-state wave function of the . What enters the analytical expressions for the form factors of the bare nucleon is its 3q bound-state wave function. Instead of solving the bound-state problem for a particular confinement potential, we rather use a simple model for this wave function, i.e. (k|) = N [(X, c i )2 + P2] Y , with k and cui denoting the quark momenta and energies in the rest frame of the nucleon. The same wave function has been used in a corresponding front-form calculation [6], from which we also take the values of the parameters p, y for later comparison. The normalization N has to be fixed such that the whole nucleon wave function, including the 3q+n component, is normalized to one. Unlike the authors of Ref. [6], who took a phenomenological nN N vertex form factor, we have calculated both, the electromagnetic form factors of the bare nucleon as well as the strong nN N vertex form factor with the same microscopic input, namely the 3q bound-state wave function O^.
With the model sketched above we achieve good agreement with the experimental data for proton electric and magnetic form factors (see Fig. 2). Our neutron magnetic form factor is also in reasonable agreement with the corresponding experimental data, the reproduction of the neutron electric form factor seems to be less satisfactory. But here one has to notice that it is a rather small quantity and the error bars on the experimental data points are, in general, large. The size of the pionic contribution to all the nucleon form factors is comparable with the one found in Ref. [6]. A significant effect of the 3q+n component on the form factors is only observed for momentum transfers Q2 < 0.5 GeV2, where it leads to a welcome modification of the Q2-dependence.
Improvements of the model can be made in several directions:
i) Take a more sophisticated 3q wave function of the (bare) nucleon, containing, e.g., a mixed SU(6) spin-flavor-symmetry component like in Ref. [6].
ii) Replace the pseudoscalar by the pseudovector nN N coupling, which guarantees correct properties in the chiral limit.
The pion-cloud contribution to the electromagnetic nucleón structure
39
q2 q2
Fig. 2. The proton electric (left) and magnetic (right) form factors as predicted by our model (solid line). The 3q valence contribution is indicated by the dotted line, the contribution due to the 3q+n non-valence component by the dashed line. The shaded area (which is hardly visible) is a parameterization of the experimental data (including uncertainties) [7].
iii) Account for other baryons, different from the nucleon, within the loop, the lightest and most important of them being the A.
The ultimate goal should, of course, be a consistent description of the baryon spectrum and the structure of the baryons. This means that one should not start with a model for the nucleon wave function, but rather with a 3q+3qn hybrid model and fit the parameters of the confinement potential and the nqq coupling strength to the baryon mass spectrum. This would give us the masses and wave functions of the (bare) baryons which are required as an input for the calculation of the strong and electromagnetic form factors of the baryons.
Acknowledgment: D. Kupelwieser acknowledges the support of the "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich" (FWF DK W1203-N16).
References
1. E. P. Biernat, W. Schweiger, K. Fuchsberger and W. H. Klink, Phys. Rev. C 79, 055203 (2009).
2. E. P. Biernat, W. H. Klink and W. Schweiger, Few Body Syst. 49,149 (2011).
3. M. Gomez-Rocha and W. Schweiger, Phys. Rev. D 86, 053010 (2012).
4. E. P. Biernat and W. Schweiger, Phys. Rev. C 89, 055205 (2014).
5. B. Bakamjian and L. H. Thomas, Phys. Rev. 92 (1953) 1300.
6. B. Pasquini and S. Boffi, Phys. Rev. D 76, 074011 (2007).
7. A. J. R. Puckett [GEp-III Collaboration], in Exclusive Reactions at High Momentum Transfer IV ed. by A. Radyushkin, World Scientific, 222 (2011); arXiv:1008.0855 [nucl-ex].
Bled Workshops in Physics Vol. 16, No. 1 p. 40
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Partial wave analysis of n photoproduction data with analyticity constraints*
M. Hadzimehmedovica, V. Kashevarovc, K. Nikonovc, R. Omerovica, H. Osmanovica, M. Ostrickc, J. Stahova, A. Svarcb, L. Tiatorc
a University of Tuzla, Faculty of Science, Bosnia and Herzegovina b Rudjer Boskovic Institute, Zagreb, Croatia
c Institut fur Kernphysik, Johannes Gutenberg Universtat Mainz, Germany
Abstract. We perform partial wave analysis of the n photoproduction on data. The obtained multipoles are consistent with the fixed-t analyticity and fixed-s analyticity. A fixed-t analyticity is imposed using Pietarinen expansion method. The invariant amplitudes obey the required crossing symmetry.
1 Introduction
A big problem in partial wave analyses are ambiguities of partial wave solutions. More than one set of partial waves describe equally well the experimental data. A first attempt to solve this problem was to require smoothness of partial waves as a function of energy. It was shown that this criteria was not enough to achieve a unique partial wave solution [1]. Furthermore, it was shown that more stringent constraints, based on the analytic properties of invariant amplitudes from Mandelstam hypothesis, should be taken into consideration. An efficient method for imposing the fixed-t analyticity on invariant amplitudes was proposed by E. Pietarinen [2-5] and was used in Karlsruhe-Helsinki partial wave analysis of nN scattering data KH80 [6-8]. In our partial wave analysis of n-photoproduction data we follow main ideas from Karlsruhe-Helsinki analysis. The method consists of two separate analyses: Fixed-t amplitude analysis (FT AA) and a single energy partial wave analysis (SE PWA). The two analyses are coupled in such a way that results from one are used as a constraint in another in an iterative procedure. The resulting partial waves (multipoles) describe experimental data adequately and are consistent with fixed-t and fixed-s analyticity as well.
2 Preparing experimental data for partial wave analysis
Our data base consists of the following experimental data: — Differential cross sections at 120 energies in the range 710 MeV < Elab < 1395 MeV [9];
* Talk presented by J. Stahov
Partial wave analysis of r| photoproduction data
41
— Beam asymmetry I at 15 energies in the range 724 MeV — 1472 MeV [10];
— Target asymmetry T at 12 energies in the range 725 MeV — 1350 MeV [11];
— Double asymmetry F at 12 energies in the range 725 MeV — 1350 MeV [11].
In SE PWA experimental data are required at a predetermined set of energies. Experimental values of beam asymmetry, target asymmetry and double polarization asymmetry are interpolated to 113 energies, where data on differential cross sections are available. A spline fit method with x2/dp = 0.7 (DP-number of data points) was used. FT AA requires experimental data at predetermined set of t values. Using the same method, data previously prepared for SE PWA were shifted to 40 t values in the range t G [—1.00 GeV2, —0.05 GeV2].
3 Fixed-t amplitude analysis
Following definition in Ref. [12], in description of n-meson photoproduction, we use crossing symmetric invariant amplitudes Bi, B2, B6, and B8/v. For a given value of variable t amplitudes are represented by two Pietarinen expansions in the form
N
Fk(v2,t) = FkN(v2,t) + (1 + zi)Y_ bu ' 4 +(1 + Z2)Y_ b2i 4, (1)
i=1 i=1
where Fk stands for invariant amplitudes Bk. FkN are explicitly known nucleon pole contributions and s,u and v = (s — u)/4m with the proton mass m are Mandelstam variables. The conformal variables z1 and z2 are defined as
a1 — V Vth1 — v2 a2 — y vth.2 — V
zi = - , , zt = - , . (2)
a1 +\l vth1 — v2 a2 + ^ vth.2 — v2
vth1 and vth2 correspond to the n and n photoproduction thresholds (yp —> n0p and yp —> np). N1 and N2 are number of parameters in expansion (1) (in our applications N1, N2 « 15). a1 and a2 are parameters which determine distribution of points on a unit circle (|z11 = |z2| = 1 ). Coefficients b1k) and b2k) in expansion (1) are determined by minimizing a quadratic form
X2 = xiata + Xpw + (3)
The term xiata is the standard exspression containing all the data at a fixed-t value
Xiata = (Dn'P(v2,t'— D"'(v2,t))2 , (4)
D n=1 Dn
where D stands for measurable quantities (ct0 = da/dH, ct0 • T, ct0 • F, ct0 • I). The sum goes over all N D available experimental values of measured quantities D for a given t value. D^ are predicted values in terms of coefficients in expansion (1).
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J. Stahov et al.
A second term xpW is also a usual x2 expression containing as "data" the helicity amplitudes calculated from the partial wave solution
x2pw = q L z (iReHkiMt'v2) - Re HPW (t,v2)]2 k=1 i=1 I
(£r)2
ki
+
[Im Hkit (t,v2) - Im H?W (t,v2)] (£i)ki
(5)
In the first iteration H{PW are calculated from an initial, already existing solution. In the subsequent iterations H£w are calculated from partial waves obtained in the single energy partial wave analysis (SE PWA) of the same set of experimental data. The weight factor q and errors £ki are unknown. They are adjusted in such a way that xiata ~ XpW. ® is Pietarinen's penalty function in the form
® = + ®2 + ®3 + 04,
(6)
where ®k is defined as
i=1
k = AikZ b^M (i + 1)3 + À2^ b
i=1
/k) J2i
(i + 1)3
A11, A21,..., A"14, A24 are weight factors determined according to the convergence test function method [5]. The final result of the fixed-t amplitude analysis consists of 40 sets of coefficients b1k) and b2k). The invariant amplitudes may be calculated at any c.m. energy W and scattering angle 0 in the physical region. Helicity amplitudes are used as a constraint in a SE PWA. Helicity amplitudes in terms of invariant amplitudes are given in the Appendix.
2
2
3.1 Single energy partial wave analysis
In the single energy partial wave analysis we minimize a quadratic form:
X2 = xiata + Xí=T .
(7)
Xdata is again a standard expression containing all the data at a given energy. For a given observable D, measured at ND angles 0i, contribution to the xdata reads:
(X
2 Ï data/d
= L
i=1
D
exp
(0i) - Dfit (0i)
A
xdata
Di
2.
data D
Dexp (0i) are experimental values of observable D with corresponding experimental errors ADi. Dfit (0i) are values of observable D calculated from partial waves which are parameters in the fit. The second term x2T is also a usual x2
2
Partial wave analysis of r| photoproduction data
43
expression containing as "data" the helicity amplitudes Hk from the fixed-t amplitude analysis. It has the form
Nc
xft =
k=1i=1
Re Hk (0i) - Re H™ (0i)
(£r )
ki
+
Im Hk (0i) - Im Hku (0i)
(£i)
ki
The angles 0t are calculated using the formula
cos 0i =
ti
m^ + 2kq '
cos 0i G [-1.00, +1.00],
(8)
where mn, q, and w are mass, c.m. momentum and c.m. energy of the n meson, and k is the c.m. momentum of the photon. Nc is the number of angles at which constraining amplitudes are given. Errors of real and imaginary parts (eR), (ei) are not determined. They are adjusted in such a way that xiata ~ Xft. After performing SE PWA at predetermined energies, the obtained partial wave values are used as a constraint in the fixed-t amplitude analysis. The "data" in the term xpW of (3) are to be calculated using these partial waves.
Our iterative procedure is shown in Fig 1.
Helicity amplitudes from initial solution
At each i-value perform fixed-i amplitude analysis Minimize:
X Xdata ~^~Xpw
At each of N energies perform single energy partial wave analysis Minimize:
X* = Xdata + Xft
Use results from single energy partial wave analysis to calculate helicity amplitudes which are used as a constraint in fixed-i amplitude analysis
Fig. 1. (Color online) Iterative procedure in a combined single energy partial wave analysis and fixed-t amplitude analysis.
2
2
To make our analysis easier to follow, we give more details about important steps after preparing input data as described in section 2.
1. Take an initial solution (MAID [13] or Bonn-Gatchina [14,15]) and calculate all four invariant amplitudes Bt(W, t) at all t values and energies where input data are available.
2. Perform the Pietarinen expansion for all invariant amplitudes using equation (1) with conformal variables defined in formula (2).
44
J. Stahov et al.
3. Calculate helicity amplitudes from invariant amplitudes (see Appendix).
4. For all t values perform a non-linear fit of observables minimizing the quadratic form (3). As starting values of parameters b|k) and bi,k) take coefficients obtained in step 2. Calculate term xpw using initial solution to calculate Hpw. This step completes the FT AA.
5. At a given energy W calculate helicity amplitudes Hk(W, cos9i), where cos9i are given by formula (8). Use coefficients b|k) and bi,k) from FT AA for corresponding t-values.
6. Perform a non-linear SE PWA using helicity amplitudes obtained in step 5 as a constraint. As starting values for partial waves (multipoles) use the same initial solution as in step 1.
7. Use results from step 6 in step 1 and perform next iteration. Our preliminary results show that, depending on the strength of constraints, it is enough to perform 2-3 iterations to get a stable final solution.
In Fig. 2 fits of invariant amplitudes are shown at t = -0.15 GeV2. Multipoles with L < 3, obtained after two iterations, are shown in Figs. 3 and 4. The Eta-Maid2015b solution was chosen as a starting solution in both analyses, FT AA and SE PWA.
Real part
3 2
„ 1
CM
n 0
CD
Imag part 1
0.5 0
-0.5 -1 -1.5 -2 -2.5 -3 -3.5
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
Fig. 2. (Color online) Red diamonds and blue circles show initial real and imaginary values of invariant amplitudes. As initial solution invariant amplitudes for t = -0.15 GeV2 from etaMAID2015b [13] are used. The red and blue lines show the Pietarinen fits to real and imaginary parts of invariant amplitudes, respectively
Partial wave analysis of r| photoproduction data 45
Real part
25 20 15 10 5 0 -5
-10 1450
2.5 2 1.5 1
0.5 0
-0.5
1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.25 0.2 0.15 0.1 0.05 0
-0.05 -0.1 -0.15 -0.2 -0.25
145015001550160016501700175018001850
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
W[MeV]
3 2.5 2 1.5 1
0.5 0
-0.5 -1 -1.5 -2 -2.5
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Real part
0.2 0.15 0.1 0.05 0
-0.05
-0.1
0.5 0.4 0.3
Ê a2 E 0.1
oj 0
S-0.1 -0.2 -0.3 -0.4
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.8
0.6
0.4
I 0.2
CVJ 0 LU
-0.2 -0.4 -0.6
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
0.6 0.4 0.2 I 0
cvj -0.2 M
-0.4 -0.6
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
-0.8
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Fig. 3. (Color online) Real and imaginary parts of multipoles obtained from SE PWA in 2nd iteration are shown as red diamonds and blue circles. The initial solution etaMAID2015b is given as red and blue solid lines.
46
J. Stahov et al.
Real part
0.15 0.1 m 0.05
[ 0
h >
^ -0.05 -0.1
-0.15 1450
) 1550 1600 1650 1700 1750 1800 1850 W[MeV]
1500
1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.2
0.1 F 0 -0.1 3-0.2 -0.3 -0.4
1450 1500 1550 1600 1650 1700 1750 1 W[MeV]
0.15 0.1 m] 0.05
^ 0 3-
^ -0.05 -0.1 -0.15
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Fig. 4. [Continued from previous page.] Caption as in Fig. 3.
4 Conclusions
A SE PWA with fixed-t constraints has been performed and multipoles, consistent with crossing symmetry and fixed-t analyticity, have been obtained. The helicity amplitudes from fixed-t show good consistency with fixed-s analyticity. It implies that our amplitudes are consistent with both, fixed-t and fixed-s analyticity.
Acknowledgment
This work was supported in part by the Federal Ministry of Education and Science, Bosnia and Herzegovina, Grant No. 05-39-3545-1/14 and by the Deutsche Forschungsgemeinschaft, Collaborative Research Center 1044.
Appendix
A Multipole expansion of invariant amplitudes
In partial wave analysis of pseudoscalar meson photoproduction it is convenient to work with CGLN amplitudes [16] giving simple representations in terms of
Partial wave analysis of r| photoproduction data 47 electric and magnetic multipoles and derivatives of Legendre polynomials
Ft = ¿[(IM1+ + El+)Pl'+1 (x) + ((I + 1 )Ml+ + El_)Pl'_1 (x)], 1=0
CO
F 2 = + 1)Ml+ + 1Ml-]Pl(x), l=i
CO
F3 = ^[(El+ - Ml+)Pl'+1 + (El- + Ml-)Pl-1 (x)],
(A.1)
l=i
F4 = - El+ - Ml- - El-]Pl"(x).
l=2
Another common set of amplitudes are helicity amplitudes, which are linearly related to the CGLN amplitudes
1 9
Hi = : sin 9cos 9(F3 + F4),
H2 = V2cos 2[(F2 - Fi ) + 1 ;Ços 9 (F3 - F4)],
1 9
H3 = — sin 9 sin 9 (F3 - F4),
H4 = V2sin 2[(Fi + F2)+ 1 + ;Ços 9 (F3 + F4)]. The relations between CGLN and invariant amplitudes are given by
(A.2)
/FA /BA
F2 = M • B2
F3 B6
VF4/ VBs/
(A.3)
with the matrix M:
M =
1
2W(s - m2)
(s-m2) (s-m2) 0 (t-m2 )(m-W) 2as7 (t-m^n) 0 (t-mn)(m+W) 2a4 (t-m2)
ai 0 2(m+W) a2 0 2(m-W)
ai (m+W) a2 (m-W) a3 (s-u) a4 (s-u)
ai a2 2a3 2a4
and
ai
a2 =
V(Ei + m)(E2 + m) 8nW
V(Ei - m)(E2 - m) 8nW
(A.4)
V(Ei - m)(E2 - m)(E2 + m)
a3 =-8nw-=a2 •(E2+m)
48
J. Stahov et al.
y^Ei + m)(E2 + m)(E2 - m) =-8nW-= ai ^ (E2 - m)
s +1 + u = ^ = 2m2 +
2 s — u
m_, v = —-
4m
where E1 and E2 are c.m. energies of the incoming and outgoing nucleons and W is the total c.m. energy.
References
1. J. E. Bowcock and H. Burkhardt, Rep. Prog. Phys. 38 1099 (1975).
2. E. Pietarinen, Nucl. Phys. B 49 315 (1972).
3. E. Pietarinen, Nucl. Phys. B 55, 541 (1973).
4. E. Pietarinen, Nucl. Phys. B 107, 21 (1976).
5. E. Pietarinen, Nuovo Cim. 12A 522 (1972).
6. G. Hohler, Pion Nucleon Scattering, Part 2, Landolt-Bornstein: Elastic and Charge Exchange Scattering of Elementary Particles, Vol. 9b (Springer-Verlag, Berlin, 1983).
7. G. Hohler, F. Kaiser, R. Koch, E. Pietarinen, Physik Daten 12N1 1 (1979).
8. R. Koch, E. Pietarinen, Nucl. Phys. A 336, 331 (1980).
9. E. F. McNicoll et al. (Crystal Ball Collaboration at MAMI), Phys. Rev. C 82, 035208 (2010).
10. O. Bartalini at al., Eur. Phys. J. A 33 169 (2007).
11. C.S. Akondi et al. (A2 Collaboration at MAMI) Phys. Rev. Lett. 113,102001 (2014).
12. I. G. Aznauryan, Phys. Rev. C 67, 015209 (2003).
13. V. Kashevarov, Proceedings from Mini-Workshop Bled 2015.
14. A.V. Anisovich, R. Beck, E. Klempt, V.A. Nikonov, A.V. Sarantsev, and U. Thoma, Eur. Phys. J. A 48 15 (2012).
15. A.V. Anisovich, E. Klempt, V.A. Nikonov, A.V. Sarantsev, U. Thoma, Eur. Phys. J. A 47 153 (2011).
16. G. F. Chew, M. L. Goldberger, F. E. Low, Y. Nambu, Phys. Rev. 106 1345 (1957).
Bled Workshops in Physics Vol. 16, No. 1 p. 49
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Progress in Neutron Couplings*
W. J. Briscoe and I. Strakovsky
The George Washington University, Washington, DC 20052, USA
Abstract. An overview of the GW SAID group effort to analyze pion photoproduction on the neutron-target will be given. The disentanglement the isoscalar and isovector EM couplings of N* and A* resonances does require compatible data on both proton and neutron targets. The final-state interaction plays a critical role in the state-of-the-art analysis in extraction of the yu —> nN data from the deuteron target experiments. It is important component of the current JLab, MAMI-C, SPring-8, CBELSA, and ELPH programs.
1 Introduction
The N* family of nucleon resonances has many well established members [1], several of which exhibit overlapping resonances with very similar masses and widths but with different JP spin-parity values. Apart from the N(1535) 1/2- state, the known proton and neutron photo-decay amplitudes have been determined from analyses of single-pion photoproduction. The present work reviews the region from the threshold to the upper limit of the SAID analyses, which is CM energy W = 2.5 GeV. There are two closely spaced states above A(1232)3/2+: N(1520)3/2- and N(1535)1/2-. Up to W - 1800 MeV, this region also encompasses a sequence of six overlapping states: N(1650)1/2-, N(1675)5/2-, N(1680)5/2+, N(1700)3/2-, N(1710)1/2+, and N(1720)3/2+.
One critical issue in the study of meson photoproduction on the nucleon comes from isospin. While isospin can change at the photon vertex, it must be conserved at the final hadronic vertex. Only with good data on both proton and neutron targets can one hope to disentangle the isoscalar and isovector electromagnetic (EM) couplings of the various N* and A* resonances (see Refs. [2]), as well as the isospin properties of the non-resonant background amplitudes. The lack of yn —» n-p and yn —» n0n data does not allow us to be as confident about the determination of neutron EM couplings relative to those of the proton. For instance, the uncertainties of neutral EM couplings of 4* low-lying N* resonances, A(nAi/2) vary between 25 and 140% while charged EM couplings, A(pA1/2), vary between 7 and 42%. Some of the N* baryons [N(1675)5/2-, for instance] have stronger EM couplings to the neutron relative to the proton, but the parameters are very uncertain [1]. One more unresolved issue relates to the second P11, N(1710) 1/2+. That is not seen in the recent nN partial-wave analysis
* Talk presented by I. Strakovsky
50 W. J. Briscoe and I. Strakovsky
(PWA) [3], contrary to other PWAs used by the PDG14 [1]. A recent brief review of its status is given in Ref. [4].
Additionally, incoherent pion photoproduction on the deuteron is interesting in various aspects of nuclear physics, and particularly, provides information on the elementary reaction on the neutron, i.e., yn —» nN. Final-state interaction (FSI) plays a critical role in the state-of-the-art analysis of the yn —» nN interaction as extracted from yd —» nNN measurements. The FSI was first considered in Refs. [5] as responsible for the near-threshold enhancement (Migdal-Watson effect) in the NN mass spectrum of the meson production reaction NN NNx. In Ref. [6], the FSI amplitude was studied in detail.
2 Complete Experiment in Pion Photoproduction
Originally, PWA arose as the technology to determine amplitude of the reaction via fitting scattering data. That is a non-trivial mathematical problem - loking for a solution of ill-posed problem following to Hadamard, Tikhonov et al. Resonances appeared as a by-product (bound states objects with definite quantum numbers, mass, lifetime and so on).
There are 4 independent invariant amplitudes for a single pion photoproduction. In order to determine the pion photoproduction amplitude, one has to carry out 8 independent measurements at fixed (s, t) (the extra observable is necessary to eliminate a sign ambiguity).
There are 16 non-redundant observables and they are not completely independent from each other, namely 1 unpolarized, da/dH; 3 single polarized, I, T, and P; 12 double polarized, E, F, G, H, Cx, Cz, Ox, Oz, Lx, Lz, Tx, and Tz measurements. Additionally, there are 18 triple-polarization asymmetries [9 (9) for linear (circular) polarized beam and 13 of them are non-vanishing] [7]. Obviously, the triple-polarization experiments are not really necessary from the theoretical point of view while such measurements will play a critical role to keep systematics under control.
3 Neutron Database
Experimental data for neutron-target photoreactions are much less abundant than those utilizing a proton target, constituting only about 15% of the present worldwide known GW SAID database [8]. The existing yn —» n-p database contains mainly differential cross sections and 15% of which are from polarized measurements. At low to intermediate energies, this lack of neutron-target data is partially compensated by experiments using pion beams, e.g., n-p —» yn, as has been measured, for example, by the Crystal Ball Collaboration at BNL [9] for the inverse photon energy E = 285 - 689 MeV and 0 = 41° — 148°, where 0 is the inverse production angle of n- in the CM frame. This process is free from complications associated with the deuteron target. However, the disadvantage of using the reaction n-p —» yn is the 5 to 500 times larger cross sections for n-p —» yn —» ynn,
Progress in Neutron Couplings
51
1050 1350 1450 1650 1850 2050 2350 2450 ¥ (MeV)
1050 1250 1450 1650 1B50 2050 2250 3450 1050 1250 1450 1650 1850 2050 3250 2450
¥ (MeV)
¥ (MeV)
7n->7T°n
1050 1250 1450 1650 1850 2050 2250 2450 ¥ (MeV)
1050 1250 1450 1650 1850 2050 2250 2450 ¥ (MeV)
1050 1250 1450 1650 1850 2050 2250 2450 ¥ (MeV)
Fig. 1. Data available for single pion photoproduction of the neutron as a function of CM energy W [8]. The number of data points, dp, is given in the upper right hand side of each subplot. Top panel: The first subplot (blue) shows the total amount of yn —> n-p data available for all observables, the second subplot (red) shows the amount of da/dO, data available, the third subplot (green) shows the amount of P observables data available. Bottom panel: The first subplot (blue) shows the total amount of yn —> n0n data available for all observables, the second subplot (red) shows the amount of da/dO data available, the third subplot (green) shows the amount of P observables data available.
depending on E and 0, which causes a large background, and there were no tagging high flux pion beams.
Figure 1 summarizes the available data for single pion photoproduction on the neutron below W = 2.5 GeV. Some high-precision data for the yn —» n-p and yn —} n0n reactions have been measured recently. We applied our GW-ITEP FSI corrections, covering a broad energy range up to E = 2.7 GeV [6], to the CLAS and A2 Collaboration yd —» n-pp measurements to get elementary cross sections for yn —» n-p [10,11]. In particular, the new CLAS cross sections have quadrupled the world database for yn —» n-p above E = 1 GeV. The FSI correction factor for the CLAS (E = 1050 - 2700 MeV and 0 = 32° - 157°) and MAMI (E = 301 -455 MeV and 0 = 45° - 125°) kinematics was found to be small, Act/ct < 10%.
Obviously, that is not enough to have compatible proton and neutron databases, specifically the energy binning of the CLAS measurements is 50 MeV or, in the worst case, 100 MeV while A2 Collaboration measurements are able to have 2 to 4 MeV binning. The forward direction, which is doable for A2 vs. CLAS, is critical for evaluation of our FSI treatment.
4 Neutron Data from Deuteron Measurements
The determination of the yd —» n- pp differential cross sections with the FSI, taken into account (including all key diagrams in Fig. 2), were done, as we did recently [6,10,11], for the CLAS [10] and MAMI data [11]. The SAID of GW Data Analysis Center (DAC) phenomenological amplitudes for yN —» nN [12], NN —} NN [13], and nN —» nN [3] were used as inputs to calculate the diagrams
52 W. J. Briscoe and I. Strakovsky
in Fig. 2. The Bonn potential (full model) [14] was used for the deuteron description. In Refs. [10,11], we calculated the FSI correction factor R(E,9) dependent on photon energy, E, and pion production angle in CM frame 0 and fitted recent CLAS and MAMI da/dH versus the world yN —» nN database [8] to get new neutron multipoles and determine neutron resonance EM couplings [10].
Fig. 2. Feynman diagrams for the leading components of the yd —> n-pp amplitude. (a) Impulse approximation (IA), (b) pp-FSI, and (c) nN-FSI. Filled black circles show FSI vertices. Wavy, dashed, solid, and double lines correspond to the photons, pions, nucleons, and deuterons, respectively.
Results of calculations and comparison with the experimental data on the differential cross sections, dayd/dH, where H and 0 are solid and polar angles of outgoing n- in the laboratory frame, respectively, with z-axis along the photon beam for the reaction yd —» n-pp are given in Fig. 3 for a number of the photon energies, E.
The FSI corrections for the CLAS and MAMI quasi-free kinematics were found to be small, as mentioned above. As an illustration, Fig. 4 shows the FSI correction factor R(E, 0) = (da/dnnp)/(daIA/dnnp) for the yn —» n-p differential cross sections as a function of the pion production angle in the CM (n — p) frame, 0, for different energies over the range of the CLAS and MAMI experiments. Overall, the FSI correction factor R(E, 0) < 1, while the effect, i.e., the (1 -R) value, vary from 10% to 30%, depending on the kinematics, and the behavior is very smooth versus pion production angle. We found a sizeable FSI-effect from S-wave part of pp-FSI at small angles. A small but systematic effect |R — 11 << 1 is found in the large angular region, where it can be estimated in the Glauber approach, except for narrow regions close to 0 ~ 0° or 0 ~ 180°. The yn —» n0n case is much more complicate vs. yn —» n-p because n0n final state can come from both yn and yp initial interactions [16]. The leading diagrams for yd —» n0pn are similar as given on Fig. 2.
5 New Neutron Amplitudes and neutron EM Couplings
The solution, SAID GB12 [10], uses the same fitting form as SAID recent SN11 solution [17], which incorporated the neutron-target CLAS da/dH for yn —» n-p [10] and GRAAL Is for both yn -> n-p and yn -> n0n [18,19] (Fig. 5). This fit form was motivated by a multichannel K-matrix approach, with an added phenomenological term proportional to the nN reaction cross section. However, these new CLAS cross sections departed significantly from our predictions at the
Progress in Neutron Couplings
53
Fig. 3. The differential cross section, dayd/dO, of the reaction yd —> n-pp in the laboratory frame at different values of the photon laboratory energy E < 1900 MeV; 0 is the polar angle of the outgoing n-. Dotted curves show the contributions from the IA amplitude [Fig. 2(a)]. Successive addition of the NN-FSI [Fig. 2(b)] and nN-FSI [Fig. 2(c)] amplitudes leads to dashed and solid curves, respectively. The filled circles are the data from DESY bubble chamber [15].
higher energies, and greatly modified PWA result [10] (Fig. 5). Recently, the BnGa group reported a neutron EM coupling determination [21] using the CLAS Collaboration yn —» n-p because n0n final state can come from both yn and yp initial interact da/dn with our FSI [10] (Table 1). BnGa13 and SAID GB12 used the same (almost) data [10] to fit them while BnGa13 has several new Ad-hoc resonances.
Overall: the difference between MAID07 with BnGa13 and SAID GB12 is rather small but resonances may be essentially different (Table 1). The new BnGa13 [21] has some difference vs. GB12 [10], PDG14 [1], for instance, for N(1535)1/2-, N(1650)1/2-, and N(1680)5/2+.
54 W. J. Briscoe and I. Strakovsky
Fig. 4. The correction factor R(E,0), where 0 is the polar angle of the outgoing n- in the rest frame of the pair fast proton. The kinematic cut, Pp > 200 MeV/c, is applied. The solid (dashed) curves are obtained with both nN- and NN-FSI (only NN-FSI) taken into account.
6 Work in Progress
At MAMI in March of 2013, we collected deuteron data below E = 800 MeV with 4 MeV energy binning [23] and will have a new experiment below E = 1600 MeV [24] in the fall of 2016.
The experimental setup provides close to 4n sr coverage for outgoing particles. The photons from n0 decays and charged particles are detected by the CB and TAPS detection system. The energy deposited by charged particles in CB and TAPS is, for the most part, proportional to their kinetic energy, unless they punch through crystals of the spectrometers. Clusters from the final-state neutrons provide information only on their angles. Separation of clusters from neutral particles and charged ones is based on the information from MWPC, PID, and TAPS veto. Separation of positive and negative pions can be based on the identification of the final-state nucleon as either a neutron or a proton. Since cluster energies from charged pions are proportional to their kinetic energy (unless their punch through the crystals), the energy of those clusters can be very low close to reaction threshold.
Progress in Neutron Couplings
55
Table 1. Neutron helicity amplitudes A1/2 and A3/2 (in [(GeV)-1/2 x 10-3] units) from the SAID GB12 [10] (first row), previous SAID SN11 [17] (second row), recent BnGa13 by the Bonn-Gatchina group [21] (third row), recent Kent12 by the Kent State Univ. group [22] (forth row), and average values from the PDG14 [1] (fifth row).
Resonance nA i/2 Resonance n-di/2 1^3/2 Ref.
JV(1535)l/2" —58± 6 JV(1520)3/2" —46± 6 —115± 5 SAID GB12
60+ 3 —47± 2 —125± 2 SAID SN11
—93±11 —49± 8 — 113±12 BnGal3
—49± 3 —38± 3 -101±4 Kent 12
—46±27 — 59+ 9 -139±11 PDG14
N( 1650)1/2" —40±10 JV(1675)5/2" — 58± 2 -80±5 SAID GB 12
—26± S —42± 2 -60±2 SAID SN11
25±20 —60± 7 -S8±10 BnGal3
11±2 —40± 4 —68± 4 Kent 12
— 15±21 —43±12 -S8±13 PDG14
JV(L440)l/2+ 4S±4 JV(1680)5/2+ 26± 4 -29±2 SAID GB 12
45±15 50± 4 -47±2 SAID SN11
43±12 34±6 -44±9 BnGa 13
40± 5 29± 2 —59± 2 Kent 12
40±10 29±10 -33±9 PDG 14
Monte Carlo simulations, which tracks reaction products through a realistic model of the detector system together with the reconstruction program, is used to calculate acceptance to various channels. So to detect the reactions under study with our setup, we have to take data with almost open trigger. Acceptance for reaction yn —» n0n varies from 70% at 0.8 GeV to 30% at 1.5 GeV of the incident-photon energy. Acceptance of reaction yp —» n+n drops at higher beam energies as charged pions punch through the crystals, and the energy of the neutron cluster does not reflect its kinetic energy. Reaction yn —» n-p above 0.8 GeV has an acceptance that is better than that for yp —» n+n as the energy and angles of the cluster from the outgoing proton can be used to reconstruct the reaction kinematics.
We are going to use our FSI technology to apply for the upcoming JLab CLAS (g13 run period) da/da for yn -» n-p covering E = 400 - 2500 MeV and 9 = 18° -152° [25]. This data set will bring about 11k new measurements which quadruple the world yn —» n-p database. The ELPH facility at Tohoku Univ. will bring new da/da for yn -> n0n below E = 1200 MeV [26].
7 Summary for Neutron Study
• The differential cross section for the processes yn —» n-p was extracted from new CLAS and MAMI-B measurements accounting for Fermi motion effects in the IA as well as NN- and nN-FSI effects beyond the IA.
• Consequential calculations of the FSI corrections, as developed by the GW-ITEP Collaboration, was applied.
• New cross sections departed significantly from our predictions, at the higher energies, and greatly modified the fit result.
56 W. J. Briscoe and I. Strakovsky
Fig. 5. Samples of neutron multipoles I = 1/2 and 3/2. Solid (dash-dotted) lines correspond to the SAID GB12 [10] (SN11 [17]) solution. Thick solid (dashed) lines give SAID GZ12 [10] solution (MAID07 [20]). Vertical arrows indicate mass (WR), and horizontal bars show full, r, and partial, rnN, widths of resonances extracted by the Breit-Wigner fit of the nN data associated with the SAID solution WI08 [3].
New yn —» n-p and Yn —!> n0n data will provide a critical constraint on the determination of the multipoles and EM couplings of low-lying baryon resonances using the PWA and coupled channel techniques.
Progress in Neutron Couplings
57
• Polarized measurements at JLab/JLab12, MAMI, SPring-8, CBELSA, and ELPH will help to bring more physics in.
• FSI corrections need to apply.
Acknowledgements
The authors are grateful to A. E. Kudryavtsev, V. V. Kulikov, M. Maremianov,
V. E. Tarasov, and R. L. Workman for many useful communications and discussions. This material is based upon work supported by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-
FG02-99-ER41110.
References
1. K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014).
2. K.M. Watson, Phys. Rev. 95, 228 (1954); R.L. Walker, Phys. Rev. 182,1729 (1969).
3. R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, Phys. Rev. C 74, 045205 (2006).
4. Ya.I. Azimov and I.I. Strakovsky, Proceedings of the XVth International Conference on Hadron Spectroscopy (Hadron 2013), Nara, Japan, Nov. 2013, PoS (Hadron 2014) 034.
5. A.B. Migdal, JETP 1, 2 (1955); K.M. Watson, Phys. Rev. 88,1163 (1952).
6. V.E. Tarasov, W.J. Briscoe, H. Gao, A.E. Kudryavtsev, and I.I. Strakovsky, Phys. Rev. C 84, 035203 (2011).
7. A.M. Sandorfi, B. Dey, A. Sarantsev, L. Tiator, and R. Workman, AIP Conf. Proc. 1432, 219 (2012); K. Nakayama, private communication, 2014.
8. W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, Institute of Nuclear Studies of The George Washington University Database; http://gwdac.phys.gwu.edu/analysis/pr_analysis.html.
9. A. Shafi et al, Phys. Rev. C 70, 035204 (2004).
10. W. Chen, H. Gao, W.J. Briscoe, D. Dutta, A.E. Kudryavtsev, M. Mirazita, M.W. Paris, P. Rossi, S. Stepanyan, I.I. Strakovsky, V.E. Tarasov, and R.L. Workman, Phys. Rev. C 86, 015206 (2012).
11. W.J. Briscoe, A.E. Kudryavtsev, P. Pedroni, I.I. Strakovsky, V.E. Tarasov, and R.L. Workman, Phys. Rev. C 86, 065207 (2012).
12. M. Dugger, J.P. Ball, P. Collins, E. Pasyuk, B.G. Ritchie, R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, R.L. Workman et al. (CLAS Collaboration), Phys. Rev. C 76, 025211 (2007).
13. R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, Phys. Rev. C 76, 025209 (2007).
14. R. Machleidt, K. Holinde, and C. Elster, Phys. Rep. 149,1 (1987).
15. P. Benz et al. (Aachen-Bonn-Hamburg-Heidelberg-Muenchen Collaboration), Nucl. Phys. B 65,158 (1973).
16. V. Tarasov, A. Kudryavtsev, W. Briscoe, M. Dieterle, B. Krusche, I. Strakovsky, and M. Ostrick, to be published in Yad. Fiz. 79 (2016) [Phys. At. Nucl. 79 (2016)] ; arXiv:1503.06671 [hep.ph].
17. R.L. Workman, W.J. Briscoe, M.W. Paris, and I.I. Strakovsky, Phys. Rev. C 85, 025201 (2012).
18. G. Mandaglio et al. (GRAAL Collaboration), Phys. Rev. C 82, 045209 (2010).
58 W. J. Briscoe and I. Strakovsky
19. R. Di Salvo et al. (GRAAL Collaboration), Eur. Phys. J. A 42,151 (2009).
20. D. Drechsel, S.S. Kamalov, and L. Tiator, Eur. Phys. J. A 34, 69 (2007); http://www.kph.uni-mainz.de/MAID/ .
21. A. Anisovich, V. Burkert, E. Klempt, V.A. Nikonov, A.V. Sarantsev, and U. Thoma, Eur. Phys. J. A 49, 67 (2013).
22. M. Shrestha and D.M. Manley, Phys. Rev. C 86, 055203 (2012).
23. Meson production off the deuteron. I, Spokespersons: W.J. Briscoe and I.I. Strakovsky (A2 Collaboration), MAMI Proposal MAMI-A2-02/12, Mainz, Germany, 2012.
24. Meson production off the deuteron. II, Spokespersons: W.J. Briscoe, V.V. Kulikov, K. Livingston, and I.I. Strakovsky (A2 Collaboration), MAMI Proposal MAMI-A2-02/13, Mainz, Germany, 2013.
25. P. Mattione, Proceedings of the XVth International Conference on Hadron Spectroscopy (Hadron 2013), Nara, Japan, Nov. 2013, PoS (Hadron 2014) 096.
26. T. Ishikawa et al., Proceedings of the XVth International Conference on Hadron Spectroscopy (Hadron 2013), Nara, Japan, Nov. 2013, PoS (Hadron 2014) 095.
Bled Workshops in Physics Vol. 16, No. 1 p. 59
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Exciting Baryon Resonances with Meson Photoproduction*
L. Tiatora, A. Svarcb
a Institut fur Kernphysik, Johannes Gutenberg Universtat Mainz, Germany b Rudjer Boskovic Institute, Zagreb, Croatia
Abstract. Light hadron spectroscopy is still an exciting field in nuclear and particle physics. Even 50 years after the discovery of the Roper resonance and more than 30 years after the pioneering work of Hoehler and Cutkosky many questions remain for baryon resonances. Nowadays the main excitation mechanism is photo- and electroproduction of mesons, studied at electron accelerator labs as MAMI, ELSA and JLab. In a combined effort, pole positions and residues are searched from partial waves, which are obtained in a partial wave analysis from recently measured polarization observables using analytical constraints from fixed-t dispersion relations. Special emphasis is placed on the pole structure of baryon resonances on different Riemann sheets.
1 Introduction
Fifty years ago the Roper resonance was found in partial wave analysis (PWA) of pion nucleon scattering [1]. In the following decades more than 30 N and A resonances were also found in PWA. For many of these resonances the properties are still uncertain and need to be improved in more precise experiments, which is nowadays only possible with photon and electron beams. Due to the helicity nature of the photon in the initial state, the number of invariant amplitudes is twice as large and the number of observables is a factor of four larger than in pion nucleon scattering. Therefore, a model independent determination of the partial waves and the underlying nucleon resonances is far more involved and improved analysis tools are required.
2 Resonances as poles on different Riemann sheets
Thresholds and resonance positions are commonly used as the most important and physical properties of partial waves in scattering and production reactions. However, at a closer look, resonance positions described in a Breit-Wigner ansatz appear different in different analyses, especially when also different reaction channels are analyzed. Also production thresholds, as nnN or nnnN are not the most relevant positions, where new dynamics is observed. E.g. at the nnN threshold,
* Talk presented by L. Tiator
60
L. Tiator, A. Svarc
W = mp + 2mn = 1208 MeV, no single partial wave shows a signature of inelasticity, even as this process is kinematically allowed.
The relevant properties of partial waves are the pole positions and well selected real or complex branch points (b.p.). Pole positions have long been realized as the fundamental resonance parameters that are not influenced by background contributions, which will be different for different reaction channels. In photoproduction, background is very small for n or n' production, but large or even very large for n and K production. In the latter case, the background is not very well known, even the coupling constants of the Born terms for (y, K) are quite uncertain.
Real branch points coincide with thresholds, like nN, nN, n'N, complex branch points appear as effective branch points for 3- and more-body final states. A very important complex branch point is the nA b.p. with Wbp = 135 + 1210 — 50i = (1345 — 50i) MeV and also the pN b.p. with Wbp = 763 — 72i + 938 = (1701 — 72i) MeV. These branch points play an important role in the Pn partial wave, other partial waves are also influenced by less amount. Their role is especially pronounced, if a pole position gets close to such a complex b.p., which is the case for Pn (1440) with Wp = (1365 — 95i) MeV and Pn (1710) with Wp = (1720 — 115i) MeV.
This knowledge is used in the Laurent-plus-Pietarinen expansion (L+P) of partial waves, recently developed by the Zagreb/Tuzla group and applied so far to nN scattering, pion photoproduction and coupled n,n channels [2,3]. Photoproduction of n and n' and pion electroproduction analyses are in progress.
For a given partial wave, e.g. for nN —» nN or yN —» nN, a = {L, J, T} with angular momentum L, spin J and isospin T, the partial wave amplitude can in general be split in a resonance and a background part, where the background part is simply everything, that is missing in the resonance ansatz,
ta(W) = M — Wr/2iF/2el* + b-g-(W) • In general, F, |3, ^ can be functions of W, in particular for a very simple case
r(W) = ^^^ M P„N Ftotai + {nnN, nA, nN + • • •} InN (M) W
with the pion c.m. momentum q«N(W)
= V(W2 — (mp + mn)2)(W2 — (mp — m^)2) 2W
_ -y/W — (mp + m )y/W + (mp + m„) y/W — (mp — mn ^W + (mp — m„)
= 2W •
In the latter expression four square-root branch points show up, where only the first one is in the physical region and is the most important branch point for all partial waves.
Exciting Baryon Resonances with Meson Photoproduction
61
The square-root function a/x has a branch point at x = 0 and is defined on two Riemann sheets (R.S.). Usually the branch cut (b.c.) is chosen to the left as in FORTRAN, C++ or Mathematica. However, any other direction can be freely chosen, according to the convenience of a particular application. In hadronic scattering processes it is often used to the right and an especially convenient way is a branch cut downwards along the negative imaginary axis. For all those definitions the formulas remain the same, except for the square-root function which has to be replaced accordingly by
It is important to note, that the ie term in Eq. (6) is needed to assure that the real axis (physical axis) belongs to the first Riemann sheet.
In principle it makes no difference, which angle for the branch cut is chosen. Traditionally most often it is the b.c. on the positive real axis to the right side. However, this convention often leads to confusions about the different Riemann sheets, as all sheets starting from real b.p. will overlap. Even b.c. turning into different directions at each different branch point are allowed. In the following we have chosen all branch cuts downwards, as this was once suggested by Dick Arndt [4]. In this convention, all resonances appear as poles on the lower halfplane in the first Riemann sheet. In his 'bible' [5], Hoehler defines the resonances as poles in the lower half-plane of the second Riemann sheet, and this convention, where all cuts are drawn to the right, is mostly used in the literature. However, one has to be very careful in numbering the R.S. when more than one threshold is open. Then the second R.S. is always the sheet, which is entered by a direct path from the physical axis down into the next R.S. by crossing one or more cuts. In our notation we give in addition to the somewhat arbitrary numbering also the ± signs for each branch cut, which makes the definition unique.
Generally with each new branch point the number of Riemann sheets gets doubled and of course all R.S. exist in the whole complex energy plane, also below the branch points. For a partial wave with 3 decay channels one must consider in principle 8 Riemann sheets. But less important decay channels are usually ignored in order to get the number of R.S. smaller. For the A(1232) in the P33 partial wave, it can be simplified by only two R.S., where the 2 poles in the first and second R.S. are symmetric above and below the real axis.
In Fig. 1 we show as the first non-trivial case the Roper resonance on four Riemann sheets. The Roper decays to almost 100% in nN and nA, as the effective nnN channel. Introducing a complex nA branch point leads to the very interesting situation, that the Roper pole appears very near to the b.p. in the first Riemann sheet. Another 2 poles show up in the upper half-plane of the second and third R.S. and are uninteresting. A lot of interest, however, caused the fourth pole, which is in the lower half-plane of the 4. R.S. and often this has been especially reported in mostly dynamical approaches. But certainly it is a shadow pole and
b.c. to the left, b.c. to the right, b.c. downwards.
—» i v7-(z + ie)
—» %/!%/—iz
62
L. Tiator, A. Svarc
Fig. 1. Contour plots of the absolute magnitude of the Pn partial wave in the complex energy plane W with poles of the N(1440) Pn resonance on 4 different Riemann sheets. The white vertical lines show the branch cuts originating at the real b.p. nN and the complex b.p. nA and the green horizontal line shows the physical axis on the first Riemann sheet.
from Fig. 1(d) it can be judged how big the influence of this pole could be on the physical axis in Fig. 1(a). In fact it is practically negligible.
Another interesting case is the N (1535) 1/2- resonance in the Sn partial wave. In Fig. 2 it appears in a normal scenario together with its partner N (1650) 1/2-in the lower half-plane of the first Riemann sheet, if we again draw all branch cuts downwards. As it is long known and already stressed by Hoehler, the N (1535) 1/2-pole sits very close near the nN threshold and one can clearly see its influence also in the fourth R.S. Now, by a small change of parameters, this pole can move below the nN cut and appears as a shadow pole in the fourth R.S., see Fig. 3. This scenario is realized in the Argonne-Osaka model [6], where the pole was found at Wp = (1482 - 98i) MeV, only 4 MeV below nN threshold. A shadow pole in the 4. R.S., so close to the branch point, without another counter part, certainly shows up with structure in the first R.S. and mocks a regular pole of the first R.S. However, all parameters of this shadow pole are a little bit surprising with different values compared to other PWA.
3 Complete experiments
A complete experiment is a set of measurements which is sufficient to predict all other possible experiments, provided that the measurements are free of uncer-
Exciting Baryon Resonances with Meson Photoproduction
63
Fig. 2. Poles of N(1535) and N(1650) Sn resonances on 4 different Riemann sheets. The white vertical line shows the branch cut originating at the real b.p. nN. The real b.p. nN is outside the plotted range. The green horizontal line shows the physical axis on the first Riemann sheet.
Fig. 3. Poles of N(1535) and N(1650) Sn resonances on the first and fourth Riemann sheets in an alternative scenario compared to Fig. 2. The N(1535) disappears form the first R.S. and appears as a shadow pole on the fourth R.S. Notation as in Fig. 2.
tainties. Using this maximal experimental information, the four complex CGLN, helicity or transversity amplitudes can be uniquely determined up to one overall energy- and angle-dependent phase, due to bilinear products of amplitudes for all observables. Starting in the 1970s many people studied the complete experi-
64 L. Tiator, A. Svarc
ment for pseudoscalar meson photoproduction and as a benchmark publication the work of Chiang and Tabakin [7] is considered who give tables where all possible combinations for such an experiment are given with the minimal number of 8 observables. In short, these 8 observables have to be chosen with beam, target and recoil polarization, which makes it in practise very difficult. Only in the last few years this goal has been achieved at JLab with KA photoproduction, where the recoil polarization of the outgoing hyperon is given for free, due to its weak self-analyzing decay. For pion and eta photoproduction meanwhile at Mainz, Bonn and JLab all 8 observables with beam and target polarization are measured over a wide energy region and with almost full angular coverage. Most of them are currently analyzed and some are already published. The 2 missing observables with beam-recoil double polarization have only been measured in a pilot experiment at MAMI using secondary rescattering of the outgoing proton. Only a few data points with rather limited statistics were obtained [8].
However, as it was shown by Omelaenko [9] in 1981 and recently revisited by Wunderlich et al. [10,11], a complete truncated partial wave analysis can be obtained with only 5 observables, where recoil observables can be completely avoided. Under these assumptions all partial wave amplitudes up to a finite angular momentum Lmax can be uniquely determined up to an overall energy-dependent phase. At first this looks as a paradox situation, however, in this latter case of a truncated partial wave analysis, the summation over all partial waves is never the same as it is in the first case with the full angle dependent amplitudes. And in a realistic case, even Lmax = 5 is hard to realize. Therefore, the difference should be understood in such a way, that with the complete experiment of 8 observables one gets the additional information with all partial waves beyond Lmax. Wunderlich et al. further showed that the complexity of the ambiguity structure drastically increases, when partial waves are considered beyond S and P waves, the case that Omelaenko initially studied. In such a more realistic case with D and F waves an unrealistically high precision of the observables were needed in order to find a unique solution. This can only be obtained in simulations with numerical observables obtained from a model with 10 or more significant digits. In a truncated PWA the contributions from higher partial waves can either be ignored or added from a model, e.g. from Born terms and/or Regge trajectories.
Therefore, if a truncated partial wave analysis is performed from a complete experiment with realistic pseudo data or with experimental data, multiple solutions will appear, which can not be distinguished. The envelope of such a large range of equally good solutions will then produce partial wave amplitudes with very large error bands [12]. From this observation a somewhat pessimistic view can easily arise that a model-independent PWA is simply impossible.
4 Partial wave analysis with analytical constraints
The most common way to get a stable solution for single-energy (SE) PWA is a fit constrained by an energy-dependent solution in a model-dependent approach. This has been done mostly for pion photoproduction by SAID, MAID, BnGa groups, the latter also tried this for eta photoproduction. For low and dominant
Exciting Baryon Resonances with Meson Photoproduction
65
partial waves this leads to similar solutions, but for smaller and higher partial waves all solutions will be different. Furthermore, the errors given in such SE analyses are just reflecting the statistical errors of the fitted experimental observables.
In a collaboration with groups from Mainz, Tuzla and Zagreb (MTZ) we are now analyzing data sets with analytical constraints from fixed-t dispersion relations. The method is similar to the pion nucleon PWA obtained in the 80s by Hoehler and Pietarinen and is described in detail in the contribution of Stahov to this workshop [13]. It enforces analyticity both in s and in t and in particular continuity in energy. Such constraints are based on fundamental symmetries and do not follow any model assumption. Finally, our goal of getting baryon resonance parameters in a model-independent way will be reached by analyzing the model-independent SE partial wave solutions obtained in the step before.
5 Baryon resonance analysis with the L+P method
Over the last few years Svarc et al. [2,3] have developed a very efficient resonance analysis method in order to find pole positions and residues from partial wave amplitudes over a large energy range. In this approach the most important properties of partial waves, poles and branch points are used as physical parameters and an expansion in terms of Pietarinen functions is used to describe the partial wave amplitudes over the whole energy range, giving more confidence on the obtained pole parameters of baryon resonances as with local methods like the speed-plot technique, first proposed by Hoehler.
The method is well described in articles with applications on pion nucleon scattering and pion photoproduction [2,3]. In summary, the set of equations which define the Laurent expansion + Pietarinen series method (L+P method) is given by
k (i) a
T (W) = L w-V+BL(W )
i=i i
N i N 2 N 3
BL (W) = L Cn X(W)n + L dn Y(W)n + L en Z(W)n + • • •
n=0 n=0 n=0
X(W) = a-VXP-W; Y(W) = P-^Xi-W; Z(W) = I-^R-W + ••• , 1 J a + Vxp-W J p + Jxq - W' 1 ' y + Vxr - W '
where Wi, a-1] are the complex pole positions and corresponding residues and xp,xq,xr are real or complex branch points. Usually, the first b.p. xp is used as an effective b.p. for the left-hand cuts, xq is the nN threshold and xr is an effective multi-pion branch point, which can correspond to nA, nN, or any other channel. If necessary, a fourth Pietarinen etc. can be added. cn, dn, en, a, p, y are real parameters and the number of terms N1 , N2,N3 of the Pietarinen series is typically between 10-20. In Fig. 4 we show four examples of pion photoproduction partial waves (multipoles) from MAID2007 SE solutions [14], where the L+P
66 L. Tiator, A. Svarc
method yields pole parameters consistent with PDG. Further details can be found in Ref. [3].
-2 1000
2000
1400 1600 W(MeV)
1400 1600 W(MeV)
2000
Fig. 4. L+P fit to MAID2007 SE solutions. Dashed blue, and full red lines denote real and imaginary parts of multipoles respectively.
6 Summary and conclusions
The study of baryon resonances is still an exciting field in hadron physics. While a large series of resonances are already known for a long time, in most cases only the dominant branching channels are well investigated. From still ongoing experiments at Mainz, Bonn and JLab, meson photo- and electroproduction data will be available partly with unprecedented precision and with different kind of beam, target and recoil polarization. With this large new database partial wave analyses can be obtained for various channels and more accurate and also new baryon resonance properties can be analyzed. In reactions different from nN also new resonances can be found, especially in the region W > 1.8 GeV, as it was already reported in a PWA mainly from new KA photoproduction data [15].
We would like to thank the Deutsche Forschungsgemeinschaft DFG for the support by the Collaborative Research Center 1044.
References
1. L. D. Roper, Phys. Rev. Lett. 12, 340 (1964).
Exciting Baryon Resonances with Meson Photoproduction
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2. A. Svarc, M. Hadzimehmedovic, H. Osmanovic, J. Stahov, L. Tiator and R. L. Workman, Phys. Rev. C 88, 035206 (2013).
3. A. Svarc, M. Hadzimehmedovic, H. Osmanovic, J. Stahov, L. Tiator and R. L. Workman, Phys. Rev. C 89, 065208 (2014).
4. R. A. Arndt, R. L. Workman, Z. Li et al., Phys. Rev. C 42,1853 (1990).
5. G. Hohler and H. Schopper, Berlin, Germany: Springer ( 1982) 407 P. ( Landolt-Bornstein: Numerical Data and Functional Relationships In Science and Technology. New Ser., I/9B1)
6. H. Kamano, S. X. Nakamura, T.-S. H. Lee and T. Sato, Phys. Rev. C 88, 035209 (2013).
7. W. T. Chiang and F. Tabakin, Phys. Rev. C 55, 2054 (1997).
8. M. H. Sikora et al, Phys. Rev. Lett. 112, 022501 (2014).
9. A. S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).
10. Y. Wunderlich, R. Beck and L. Tiator, Phys. Rev. C 89, 055203 (2014).
11. Y. Wunderlich, Proc. of the Mini-Workshop on Exploring Hadron Resonances, Bled (Slovenia), July 5-11, 2015, in this volume.
12. A.M. Sandorfi, S. Hoblit, H. Kamano, T.-S.H. Lee, J. Phys. G 38, 053001 (2011).
13. J. Stahov et al., Proc. of the Mini-Workshop on Exploring Hadron Resonances, Bled (Slovenia), July 5-11, 2015, in this volume.
14. D. Drechsel, S. S. Kamalov, L. Tiator, Eur. Phys. J. A 34, 69 (2007).
15. A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 48, 15 (2012).
Bled Workshops in Physics Vol. 16, No. 1 p. 68
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Complete experiments in pseudoscalar meson photoproduction
Yannick Wunderlich
Helmholtz-Institut f"ur Strahlen- und Kernphysik, Universit"at Bonn, Nussallee 14-16, 53115 Bonn, Germany
Abstract. The problem of extracting photoproduction amplitudes uniquely from so called complete experiments is discussed. This problem can be considered either for the extraction of full production amplitudes, or for the determination of multipoles. Both cases are treated briefly. Preliminary results for the fitting of multipoles, as well as the determination of their error, from recent polarization measurements in the A-region are described in more detail.
1 Introduction to the formalism
For the photoproduction of a single pseudoscalar mesons, i.e. yN —» PB, it can be shown that the most general expression for the reaction amplitude, with spin and momentum variables specified in the center of mass frame (CMS), reads (cf. the work by CGLN [1])
FCGLN = íct • t Ft + ct • t ct • k x t F2 + íct • k t • ê F3 + íct • t q • ê F4 . (1)
Each spin-momentum structure in this expansion is multiplied by a complex function depending on the total energy W and meson scattering angle 0 in the CMS. The 4 functions {Ft (W, 0) ; i = 1,..., 4} are called CGLN-amplitudes and contain all information on the dynamics of the reaction.
Since all particles in the reaction except for the meson P have spin, the preparation of the spin degrees of freedom in the initial state as well as the (generally more difficult) measurement of the polarization of the recoil baryon B facilitate the experimental determination of 16 polarization observables, summarized in Table 1. All observables are definable as asymmetries among different polarization states (see [2]). They contain the unpolarized differential cross section ct0, the three single spin observables {I, T, P} (corresponding to beam, traget and recoil polarization), as well as twelve double polarization observables which are divisible into the distinct classes of beam-target (BT), beam-recoil (BR) and target-recoil (TR) observables.
Once the equations connecting the measurable observables to the model independent production amplitudes are worked out (reference [2] contains instructions on how to do this), it becomes apparent that all of these relations can be
Complete experiments in pseudoscalar meson photoproduction
69
Table 1. The 16 polarization observables accessible in pseudoscalar meson photoproduction (for a more elaborate version of this Table, cf. [2]).
Beam Target Recoil Target + Recoil
- - - - x' y ' z' x' x' z' z'
- x y z - - - x z x z
unpolarized oo T P Tx1 Lx1 Tz1 Lz1
linearly pol. I H P G Ox ' T Oz,
circularly pol. F E Cx1 Cz1
summarized by the relation
1 4 1 = k 2 £ = k2 (F'Aa'F> ' a = l--^ (2)
The 16 real profile functions Óa, connected to the polarization observables via Óa = o0Óa, are bilinear hermitean forms in the CGLN amplitudes and can be represented by the generally complex hermitean matrices /Aa (cf. [5] for a listing of those).
A change of the basis of spin quantization for the photoproduction reaction allows for the definition of different systems of spin amplitudes. Helicity amplitudes Hi (W, 9) or transversity amplitudes bt (W, 9) are possible choices (cf. [4]). The different kinds of amplitdues are all related among each other in a linear and invertible way. Therefore, they can be seen as fully equivalent regarding their information content. The expressions for the polarization observables in the afore mentioned different systems of spin amplitudes retain the mathematical structure of equation (2), while the observables are now represented by different matrices
Óa = kl (H I ral H = k1 (b lral b> • (3)
The Fa (or Fa in case of transversity amplitudes) are a set of 16 hermitean unitary Dirac F -matrices (cf. [4,5]). They have useful properties, the exploitation of which facilitates the identification of complete experiments.
2 Complete experiments for spin amplitudes
Since photoproduction allows access to 16 polarization observables but needs 4 complex amplitudes for a model independent description (constituting just 8 real numbers), the fact can be anticipated that measuring all observables would mean an overdetermination for the problem of extracting amplitudes.
This issue has triggered investigations on so called complete experiments (cf. [3,4]), which are subsets of a minimum number of observables that allow for
70 Y. Wunderlich
a unique extraction of the amplitudes. Here one generally means unique only only up to an overall phase, since equations (2, 3) are invariant by a simultaneous rotation of all amplitudes by the same phase. Also, the complete experiment problem is first of all a precise mathematical problem disgregarding measurement uncertainties.
Chiang and Tabakin have published a solution to this problem (cf. [4]) that shall be depicted here. First of all it was noted that, using the fact that the f-matrices are an orthonormal basis of the complex 4 x 4-matrices, equation (3) can be inverted in order to yield expressions for the bilinear products
b^bj = 2 Z (ra r Oa. (4)
a
This relation allows for the determination of the moduli |bt| and relative phases ^ of the bt and therefore fully constrains them up to an overall phase. Generalizations of equation (4) for helicity and CGLN amplitudes are possible (see [5]) but shall not be quoted here.
Another important property of the f is that they imply quadratic relations among the observables known as the Fierz identities (see [4])
OaOP = ^ C^O6On, (5)
M
where C^f = (1/16) Tr [f6f afnfP].
Equations (4) and (5) are all that is needed to prove that 8 carefully chosen observables suffice in order to obtain a complete experiment ( [4]). Among those should be the unpolarized cross section and the three single polarization observables. The remaining quantities have to be picked from at least two different classes of double polarization observables, with no more than two of them from the same class. The word 'prove' means in this case that for all cases mentioned in reference [4], equation (5) was used to express the missing 8 observables in terms of the measured ones.
In practical investigations of photoproduction data, the goal is not to determine the full reaction amplitudes, but rather the partial waves, in this case called multipoles.
3 Complete experiments in a truncated partial wave analysis
The expansions of the full amplitudes Ft into multipoles are known (cf. eg. [2]). In case these expansions are truncated at some finite angular momentum quantum number £max, an approximation that is justified for reactions with supressed background contributions (eg. n0 photoproduction), then the profile functions defined in equation (2) can be arranged as a finite expansion into associated Leg-endre polynomials
2^max+ p a +"Y a
Oa (W, 9) = q Z (aoa (W) Pfa (cos 0), (6)
k=Pa
(aü£ (W) =