Blejske delavnice iz fizike Letnik 10, št. 1 Bled Workshops in Physics Vol. 10, No. 1 ISSN 1580-4992 Proceedings of the Mini-Workshop Problems in Multi-Quark States Bled, Slovenia, June 29 - July 6, 2009 Edited by Bojan Golli Mitja Rosina Simon Širca University of Ljubljana and Jozef Stefan Institute DMFA - ZALOŽNIŠTVO Ljubljana, november 2009 The Mini-Workshop Problems in Multi-Quark States was organized by Jožef Stefan Institute, Ljubljana Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana and sponsored by Slovenian Research Agency Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Organizing Committee Mitja Rosina, Bojan Golli, Simon ¡Sirca List of participants Boris Arbuzov, Moscow, arbuzov@theory .sinp. msu. ru Enrique Ruiz Arriola, Granada, earriola@ugr. es Elmar Biernat, Graz, elmar.biernat@uni-graz .at AlexBlin, Coimbra, alex@fis .uc.pt Wojtek Broniowski, Krakow, b4bronio@cyf-kr.edu.pl Ki-Seok Choi, Graz, ki . choi@uni-graz . at Veljko Dmitrašinovic, Belgrade, dmitrasin@yahoo. com Bojan Golli, Ljubljana, boj an. goll i@ij s.si Brigitte Hiller, Coimbra, brigitte@fis .uc.pt Sasha Osipov, Coimbra, alexguest@malaposta . fis .uc.pt Willi Plessas, Graz, willibald.plessas@uni-graz.at Milan Potokar, Ljubljana, Milan. Potokar@ij s.si Saša Prelovšek, Ljubljana, Sasa . Prelovsek@ij s.si Dan-Olof Riska, Helsinki, riska@mappi . helsinki . fi Mitja Rosina, Ljubljana, mitj a . rosina@ij s.si Ica Stancu, Liege, fstancu@ulg .ac.be Simon Sirca, Ljubljana, simon.sirca@fmf.uni-lj .si Masashi Wakamatsu, Osaka, wakamatu@kern.phys .sci. osaka-u. ac.jp Tomi Živko, Ljubljana, tomi . zivko@ij s.si Electronic edition http://www-f1.ijs.si/BledPub/ Contents Preface............................................................. V Nambu-Jona-Lasinio model from QCD B. A. Arbuzov....................................................... 1 Renormalization and universality of NN interactions in Chiral Quark and Soliton Models E. Ruiz Arriola and A. Calle Cordon.................................... 6 Electro-magnetic meson form-factor from a relativistic coupled-channels approach E. P. Biernat, W. Schweiger, K. Fuchsberger, and W. H. Klink............... 17 Gravitational, electromagnetic, and transition form factors of the pion Wojciech Broniowsk and Enrique Ruiz Arriola........................... 20 Axial charges of nucleon resonances Ki-Seok Choi, W. Plessas, and R.F. Wagenbrunn.......................... 28 Nucleon axial couplings and [(2,0) © (0, \)]-[(!, © (2,1)] chiral multiplet mixing V Dmitrasinovic, A. Hosaka, K. Nagata................................. 31 Quadrupole polarizabilities of the pion in the Nambu-Jona-Lasinio model B. Hiller, W. Broniowski, A. A. Osipov, A. H. Blin........................ 39 Extended NJL model with eight-quark interactions A. A. Osipov, B. Hiller, A. H. Blin, J. Moreira ............................ 44 Meson-Baryon Interaction Vertices T. Melde, L. Canton, W Plessas........................................ 48 Multi-quark configurations in the baryons D.O.Riska ......................................................... 53 Multiquark hadrons Fl.Stancu........................................................... 57 Chiral Quark Soliton Model and Nucleon Spin Structure Functions M. Wakamatsu...................................................... 62 Pion electro-production in the Roper region: K-matrix approach B. Colli, S. Sirca and M. Fiolhais....................................... 71 What have we learned from the Nambu-Jona-Lasinio model MitjaRosina ........................................................ 77 Pion electro-production in the Roper region: planned experiment at the MAMI/A1 setup S. Sirca............................................................. 81 Hadronic spectroscopy at Belle M. Bracko and T. Zivko............................................... 88 Preface Repetitio est mater studiorum, said the ancient Romans. It is now time to repeat the lessons, ideas and criticisms encountered at Bled 2009. We would like to thank you for the neat presentations summarized in these Proceedings which try to represent some flesh and spirit of our coming-together. One of the red threads was the Nambu-Jona-Lasinio model. To what extent can it be derived from QCD? The Bogolyubov compensation method does provide a link. Work was done on pion polarizabilities and inclusion of four-body forces, which not only stabilizes the vacuum but also influences the behaviour in certain phase transitions. The simpified NJL gives some insights into the large-Nc limit and in the deduction of pion scattering lengths. Low-lying baryon resonances are experiencing steady progress. The peculiar shape of the electro-production amplitudes in the Roper region can be explained by an interplay of intermediate A or ct states. Classification of resonances can be facilitated by looking at the density as a function of inter-quark distance. The renor-malization of singular potentials with one-meson exchange still pose problems. The electromagnetic form-factors of baryons, the "gravitational" form-factor of the pion, the restoration of chiral symetry and the spin structure of baryons continue to attract our attention. New resonances in the charmonium spectrum present many puzzles and model descriptions of their decay channels are not yet consistent. However, admixtures of higher configurations in baryons gain more and more credit; the excess of d over u, for example, seems to require that. The four resonances studied at Belle were presented from the tetraquark point of view. What topics should we tackle next year? We have not yet decided. There are many puzzles and secret wishes hidden in your and our minds. We are exploring the "market" and we are also awaiting your suggestions. Ljubljana, November 2009 M. Rosina B. Colli S. Scirca Workshops organized at Bled > What Comes beyond the Standard Model (June 29-July 9,1998), Vol. 0 (1999) No. 1 > Hadrons as Solitons (July 6-17,1999) > What Comes beyond the Standard Model (July 22-31,1999) > Few-Quark Problems (July 8-15, 2000), Vol. 1 (2000) No. 1 > What Comes beyond the Standard Model (July 17-31, 2000) > Statistical Mechanics of Complex Systems (August 27-September 2, 2000) > Selected Few-Body Problems in Hadronic and Atomic Physics (July 7-14,2001), Vol. 2 (2001) No. 1 > What Comes beyond the Standard Model (July 17-27,2001), Vol. 2 (2001) No. 2 > Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry > Quarks and Hadrons (July 6-13, 2002), Vol. 3 (2002) No. 3 > What Comes beyond the Standard Model (July 15-25,2002), Vol. 3 (2002) No. 4 > Effective Quark-Quark Interaction (July 7-14, 2003), Vol. 4 (2003) No. 1 > What Comes beyond the Standard Model (July 17-27, 2003), Vol. 4 (2003) Nos. 2-3 > Quark Dynamics (July 12-19, 2004), Vol. 5 (2004) No. 1 > What Comes beyond the Standard Model (July 19-29,2004), Vol. 5 (2004) No. 2 > Exciting Hadrons (July 11-18, 2005), Vol. 6 (2005) No. 1 > What Comes beyond the Standard Model (July 18-28,2005), Vol. 6 (2005) No. 2 > Progress in Quark Models (July 10-17, 2006), Vol. 7 (2006) No. 1 > What Comes beyond the Standard Model (September 16-29,2006), Vol. 7 (2006) No. 2 > Hadron Structure and Lattice QCD (July 9-16, 2007), Vol. 8 (2007) No. 1 > What Comes beyond the Standard Model (July 18-28,2007), Vol. 8 (2007) No. 2 > Few-Quark States and the Continuum (September 15-22, 2008), Vol. 9 (2008) No. 1 > What Comes beyond the Standard Model (July 15-25,2008), Vol. 9 (2008) No. 2 > Problems in Multi-Quark States (June 29-July 6, 2009), Vol. 10 (2009) No. 1 > What Comes beyond the Standard Model (July 14-24, 2009), Vol. 10 (2009) No. 2 Also published in this series > Book of Abstracts, XVIIIEuropean Conference on Few-BodyProblems in Physics, Bled, Slovenia, September 8-14,2002, Edited by Rajmund Krivec, Bojan Golli, Mitja Rosina, and Simon Sirca, Vol. 3 (2002) No. 1-2 Bled Workshops in Physics Vol. 10, No. 1 p.l Nambu-Jona-Lasinio model from QCD B. A. Arbuzov Skobeltsyn Institute of Nuclear Physics of MSU, 119992 Moscow, Russia The NJL model [1-3] proves to be effective in description of low-energy hadron physics. The model starts with effective chiral invariant Lagrangian where ^ is the light quark doublet (u, d). This interaction is non-renormalizable, so one is forced to introduce an ultraviolet cut-off A. Thus we have at least two arbitrary parameters to be adjusted by comparison with real physics. It comes out that after such adjustment (and similar procedure for the vector sector and for the s-quark terms) we obtain satisfactory description of light mesons and their low-energy interactions. However, the problem how to calculate the parameters Gt and At from the fundamental QCD was not solved for a long time. The main problem here is to find a method to obtain effective interactions from fundamental gauge interactions, e.g. QCD. There are also non-local variants of the NJL model, in which one introduces a form-factor F(qt) into the effective interaction of the type (1) instead of a cut-off A. In this case again there was no regular method to obtain this function F and one has to make an arbitrary assumption for the choice. Our goal is to formulate a regular approach, which allows to obtain a unique solution for the form-factors and other necessary quantities of the effective interactions. In particular we apply this approach to the NJL effective interaction. The approach is based on the Bogoliubov compensation principle [4,5]. The main principle of the approach is to check if an effective interaction could be generated in a chosen variant of a renormalizable theory. In previous works [6-12] the Bogoliubov compensation principle was applied to studies of spontaneous generation of effective non-local interactions in renormalizable gauge theories. In view of this one performs an "add and subtract" procedure for the effective interaction with a form-factor. Then one assumes the presence of the effective interaction in the interaction Lagrangian and the same term with the opposite sign is assigned to the newly defined free La-grangian. (1) Gi ; At ; The QCD Lagrangian with two light quarks is (u and d) L = Q (^kY^^k - Q^kY^k^ - m0i|>k^k + 9siJJkYntaA£iK^ k=1 1 , Fa Fa (2) 4 i 1 i • y-^-j Let us assume that a non-local NJL interaction is spontaneously generated in this theory. We use the Bogoliubov "add and subtract" procedure to check the assumption. We have L = Lo + Lint , 1 ^ ~ ■ ~ - G1 , ,.,T.„b„ ■ T . T .A + y (Vby^ VY^ + iKby5y^iKby5y^ - \ F0VpoV , (3) 1 4 _ pa ra _ ra pa \ /¿\ a i ^r |ir 1 0 |ir1 0 |ir i • (4) Here the notation e.g. 4> 4> 4> means the corresponding non-local vertex in the momentum space i(2n)4 G i ua (p) ua (q) ub (k) ub (t) F(p, q, k, t) 6 (p + q + k +1) , (5) where F(p, q, k, t) is a form-factor, p, q, k, t are respectively incoming momenta and a, b are isotopic indices of corresponding quarks. Let us consider expression (3) as the new free Lagrangian L0, whereas expression (4) is the new interaction Lagrangian Lint. The compensation equation demands fully connected four-fermion vertices, following from Lagrangian L0, to be zero. The equation has evidently 1. a perturbative trivial solution Gi = 0; 2. but it might also have a non-perturbative non-trivial solution, which we shall look for. In the first approximation we use the following assumptions. 1. Loop numbers 0,1, 2. For one-loop case only a trivial solution exists. 2. Procedure of linearizing over form-factor, which leads to linear integral equations. 3. Intermediate UV cut-off A, results not depending on the value of this cut-off. 4. IR cut-off at the lower limit of integration by momentum squared q2 at value m2. 5. Only the first two terms of the 1 /N expansion (N = 3). 6. We look for a solution with the following simple dependence on all four variables: U i u M + V2 + vl + vj\ F(Pi, V2, Vs, P4) = F I -^—3-1 1 . (6) Then we come to the following integral equation (see [8]) , , 3G2 U , x 3 m2 \ (G? + 6Gi G2)N A2 2 \ 6x log x 1 fx 3 rx (y2-3H2)F1(y)dy + - H 2 yFi (y) dy + H x2 — 3^' 6~ 2 Fi (y) dy + yFi (y) dy + x log x Fi (y) dy + (logy + ^F! (y)dy+ (2A2-^x) y log yFi(y) dy + yFi (y) dy 3 Ft (y) dy - - log A2 yFi (y) dy + x JH GiN Fi (y) dy) I ; n = m0 ; x = p2 ; y = q2 ; (7) 2n2 4N 2tt2 V1 + 2N ) Fi(y) dy The equation has the following solution decreasing at infinity F!(z) = C! G4°(z|l, 1 1 0, a, b) +C2 G4°(z|l, 1 b, a, 1 0, ) 1 1 2' 2/ 2' 2 + C3 G4°(z|l,0,b, a, j, I) , b 1 -VI -64u0 1 + a/1 - 64u0 (8) 4 4 where x = p2, y = q2 are respectively external momentum squared and inegra-tion momentum squared, is a Meijer G-function [13], (G2 + 6Gi G2)N P 16n4 nmni „ iai , ... , ap Gpq ( Z |bi,...,bp The constants Ct are defined by the boundary conditions 3 G2 P 8n2 2 Fi (y) dy = 0 y Fi (y) dy = 0 y2 Fi (y) dy = 0. (9) These conditions and the condition A = 0 lead to the cancellation of all terms in equation (7) being proportional to A2 and log A2. So we have the unique solution. The values of the parameter u0 and the ratio of two constants Gt are also fixed u0 = 1.92•10-8 ~ 2•10- Gi =^G2. (10) y "X "X x 8 We would draw attention to a natural appearance of a small quantity u0 .So G i and G2 are both defined in terms of m0. Thus we have the unique non-trivial solution of the compensation equation, which contains no additional parameters. It is important that the solution exists only for positive G2 and due to (10) for positive G1 as well. Now we have the non-trivial solution, which lead to the following effective Lagrangian - (xpTby5iWTbY5i|j - - -y- (^V^^TV^ + ^TVSY^^VSY^ • (11) Here g2/4n = as(q2) is the running constant depending on the momentum variable. We need this constant in the low-momenta region. We assume that in this region as (q2) may be approximated by its average value as. The possible range of values of as is from 0.40 up to 0.75. Thus we come to the effective non-local NJL interaction which we use to obtain the description of low-energy hadron physics [7,8,11]. In this way we obtain expressions for all quantities under study. Analysis shows that the optimal set of low-energy parameters corresponds to as = 0.67 and m0 = 20.3 MeV. We present a set of calculated parameters for these conditions including the quark condensate,the parameters of the a-meson as well as the parameters of p and ai -mesons: as = 0.673; m0 = 20.3 MeV; mn = 135 MeV; mff = 492 MeV; Tff = 574 MeV fn = 93 MeV; m = 295 MeV; < q q >= - (222 MeV)3 ; (244 MeV)2 Mp = 926.3 MeV(771.1 ± 0.9); rp = 159.5 MeV(149.2 ± 0.7); Ma, = 1174.8 MeV( 1230 ±40); ra, = 350MeV(250 - 600); r(a^ —> an)/rai = 0.23 (0.188 ± 0.043). G1 = 7TT7T7TT7TI '> g =3.16. The upper line here is our input, while all other quantities are calculated from these two fundamental parameters. The overall accuracy may be estimated to be on the order of 10 - 15%. The worst accuracy occurs in the value of Mp (20%). It seems that the vectors and the axials need further study. Important result: average value of as ~ 0.67 agrees with calculated low-energy as [9]. So we have consistent description of low-energy hadron physics with only one dimensional parameter, e.g. m0 or fn. References 1. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961). 2. T. Eguchi, Phys. Rev. D 14, 2755 (1976). 3. M. K. Volkov and D.Ebert, Yad. Fiz. 36,1265 (1982). 4. N. N. Bogoliubov. Physica Suppl. 26,1 (1960). 5. N. N. Bogoliubov, Quasi-averages in problems of statistical mechanics. Preprint JINR D-781, (Dubna: JINR, 1961). 6. B. A. Arbuzov, Theor. Math. Phys. 140,1205 (2004). 7. B. A. Arbuzov, Phys. Atom. Nucl. 69,1588 (2006). 8. B. A. Arbuzov, M. K. Volkov, I. V. Zaitsev. J. Mod. Phys. A, 21, 5721 (2006). 9. B.A. Arbuzov, Phys.Lett. B656, 67 (2007). 10. B.A. Arbuzov, Proc. International Seminar on Contemporary Problems of Elementary Particle Physics, Dubna, January 17-18, 2008, Dubna, JINR, 2008, p. 156. 11. B. A. Arbuzov, M. K. Volkov, I. V. Zaitsev. J. Mod. Phys. A, 24, 2415 (2009). 12. B. A. Arbuzov, Eur. Phys. Journal C 61, 51 (2009). 13. H. Bateman and A. Erdelyi, Higher transcendental functions, Vol. 1. New York, Toronto, London: McGraw-Hill, 1953. Bled Workshops in Physics Vol. 10, No. 1 P. 6 Renormalization and universality of NN interactions in Chiral Quark and Soliton Models * E. Ruiz Arrióla and A. Calle Cordón Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain Abstract. We use renormalization as a tool to extract universal features of the NN interaction in quark and soliton nucleon models, having the same long distance behaviour but different short distance components. While fine tuning conditions in the models make difficult to fit NN data, the introduction of suitable renormalization conditions supresses the short distance sensitivity. Departures from universality are equivalent to extracting information on the model nucleon structure. 1 Introduction The meson exchange picture has played a key role in the development of Nuclear Physics [1,2]. However, the traditional difficulty has been a practical need to rely on short distance information which is hardly accessible directly but becomes relevant when nucleons are placed off-shell. From a theoretical point of view this is unsatisfactory since one must face uncertainties not necessarily linked to our deficient knowledge at long distances and which are difficult to quantify. On the other hand, the purely field theoretical derivation yields potentials which present short distance singularities, thereby generating ambiguities even in the case of the widely used One Boson Exchange (OBE) potential. Consider, for instance, the venerable One Pion Exchange (OPE) NN —> NN potential which for r = 0 reads V^nnM = T1 • T2CT1 • CT2Wln(r) + Ti • T2Si2W-|n(r) , (1) where the tensor operator S12 = 3ct1 • £ct2 • £ — ct1 • ct2 has been introduced and W^(T) , W|«(r) = ^%^Y2(m7tr). (2) 3 4n 3 4n Here Y0(x) = e-x/x and Y2(x) = e-x/x(1 + 3/x + 3/x2) and fnNN = mngnNN/ (2Mn); fnNN/(4n) = 0.07388 for g„NN = 13.08. As we see, the OPE potential presents a 1 /r3 singularity, but it can be handled unambiguously mathematically and with successful deuteron phenomenology [3]. Nonetheless, the standard way out to avoid the singularities in this and the more general OBE case is to implement vertex functions for the meson-baryon-baryon coupling (mAB) in the OBE * Talk delivered by E. Ruiz Arriola potentials. This correspondins to a folding in coordinate space which in momentum space becomes the multiplicative replacement VmABÎqH VmAB(q) [jmAB^2)]' (3) where q2 = qO — q2 is the 4-momentum. Standard choices are to take form factors of the mono-pole [1] and exponential [2] parameterizations pmon 1 mNN (q2 ) = A2 m A2 CN N(q2) = exp q2 m A2 (4) fulfilling the normalization condition rmNN(m2) = 1. Due to an extreme fine-tuning of the interaction, mainly in the 1 So channel, OBE potential models have traditionally needed a too large 9o>nn to overcome the mid range attraction implying one of the largest 40%) SU(3) violations known to date. In our recent works [4-9] we discuss how this problem may be circumvented with the help of renormalization ideas which upon imposing short distance insensitivity sidestep the fine tuning problem and allow natural SU(3) values to be adopted in such a way that form factors and heavy mesons play a more marginal role. Contrarily to what one might naively think, renormalization reduces the short distance dependence provided, of course, removing the cut-off and the imposed renormalization conditions are mutually compatible operations. Of course, the extended character of the nucleon as a composite and bound state of three quarks has motivated the use of microscopic models of the nucleon to provide an understanding of the short range interaction besides describing hadronic spectroscopy; quark or soliton models endow the nucleon with its finite size and incorporate basic requirements from the Pauli principle at the quark level or as dictated by the equivalent topology [10-13]. While much effort has been invested into determining the short range interactions, there is a plethora of models and related approximations; it is not obvious what features of the model are being actually tested. In fact, NN studies set the most stringent nucleon size oscillator constant value bN = 0.518fm [13] from S-waves and deuteron properties which otherwise could be in a wider range bN = 0.4 — 0.6fm. This shows that quark models also suffer from a fine tuning problem. In this contribution we wish to focus on the common and universal patterns of the various approaches and to show how these fine tunings can be reduced to a set of renormalization conditions. 2 q 2 The relevant scales From a fundamental point of view the NN interaction should be obtained as a natural solution of the 6-q system. However, in order to describe the NN interaction it is far more convenient to study two 3-q clusters with nucleon quantum numbers, a procedure also applied in recent lattice QCD investigations of the nuclear force [14,15]. NN scattering in the elastic region corresponds to resolve distances about the minimal de Broglie wavelength associated to the first inelastic pion production threshold, NN —> NNn, and corresponds to take 2ECM = 2MN + mn yielding pCM = %/ñx^TVl^" = 360MeV which means Amin - 1 /^Mn = 0.5fm. This scale is smaller than 1 n and 2n exchange (TPE) with Compton wavelengths 1.4 and 0.7fm respectively. Other length scales in the problem are comparable and even shorter namely 1) Nucleon size, 2) Correlated meson exchanges and 3) Quark exchange effects. All these effects are of similar range and, to some extent, redundant. In a quark model the constituent quark mass is related to the Nucleon and vector meson masses through Mq = MN/Nc = MV/2 which for Nc = 3 colours gives the estimate Mq = 310 — 375MeV. Exchange effects due to e.g. One-Gluon-Exchange are ~ e-2Mq r since they correspond to the probability of finding a quark in the opposite baryon. This follows from complete Vector Meson Dominance (for a review see e.g. [16]), which for the isoscalar baryon density, Pb (r), and assuming independent particle motion yields d3 xeiqx(N|pB (x)|N) = 4n i i Mw drr2|fr(r)|2j0(qr) - (5) MV + q2 suggesting a spectroscopic factor ^(r) ~ e~ at large distances. As we have said and we will discuss below these effects are somewhat marginal but if they ought to become visible they should reflect the correct asymptotic behaviour. In the constituent quark model the CM motion can be easily extracted assuming harmonic oscillator wave functions, ^(r) ~ e-b2t2/2 [10,11,13] which yield Gaussian form factors falling off much faster than the experimental ones. Skyrme models without vector mesons yield instead topological Baryon densities Pb (r) ~ e-3mnT/r7[12] corresponding to the outer pion cloud contributions which are longest range but pressumably yield only a fraction of the radius. In any case quark-exchange looks very much like direct vector meson exchange potential which is ~ e-MvT. 3 Chiral quark soliton model Most high precision NN potentials providing x2/DOF < 1 need to incorporate universally the One-Pion-Exchange (OPE) potential (including charge symmetry breaking effects) while the shorter range is described by many and not so similarly looking interactions [17]. This is probably a confirmation that chiral symmetry is spontaneously broken at longer distances than confinement, since hadronization has already taken place. It also suggests that in a quark model aiming at describing NN interactions the pion must be effectively included. Chi-ral quark models accomplish this explicitly under the assumption that confinement is not crucial for the binding of n, N and A. Pure quark models including confinement or not have to face in addition the problem of recovering the pion from quark-gluon dynamics. In between, hybrid models have become practical and popular [10,11,13]. As mentioned, all these scales around the confinement scale are mixed up. Because these effects are least understood and trigger side effects such as spurious colour Van der Waals forces arising from Hidden color singlet states [88]a states [18,19] in the (presumably doubtful) adiabatic approximation, we will cavalierly ignore the difficulties by remaining in a regime where confinement is not expected to play a role and stay with standard chiral quark models. While both the constituent chiral quark model and the Skyrme soliton model look very disparate the Chiral Quark Soliton Model embeds both models in the small and the large soliton limit respectively 1. We analyze the intuitive non-relativistic chiral quark model (NRCQM) explicitly and comment on the soliton case where similar patterns emerge. The comparison stresses common aspects of the quark soliton model pictures which could be true features of QCD. While the long distance universality between both NRCQM and Skyrme soliton model NN calculations may appear somewhat surprising this is actually so because in a large Nc framework both models are just different realizations of the contracted spin-flavour symmetry [23]. 4 The non-relativistic chiral quark model To fix ideas it is instructive to consider the chiral-quark model which corresponds to the Gell-Mann-Levy sigma model Lagrangean at the quark level [24] (the nonlinear version suggested in Ref. [25] will be discussed below), £ = q(ij8-g7tqq(ff + iY5T-7t))q + - [(9pct)2 + (9p7t)2] -U(ct,tt) (6) where U(a,n) = A2(ct2 + n2 — v2)2/8 — f^m^a is the standard Mexican hat potential implementing both spontaneous breaking of chiral symmetry as well as PCAC yielding the Goldberger-Treiman relation Mq = gnqqfn = g2qqfn at the constituent quark level. When this model is interpreted from a gradient expansion of the NJL model quarks are regarded as valence quarks whereas kinetic meson terms arise from the polarization of the Dirac sea and m2 = 4Mq + m^, which for Mq = MN/3 = MV/2 yields m2 = 650 — 770MeV. In the heavy constituent quarks limit the model implies 1 n and 1 a exchange potentials, Vlq ' (r) qq Vlq ' (r) = g 4M2 q d3p J2n)3 ,ipr (aq • p)(aq' • p) 52 + m2 qq nqq d3p ,ip r_ 1 g nqq 4n (2n)3 p2 + m2 whence baryon properties can be obtained by solving the Hamiltonian (7) H = Z i=1 p? 2MC + Mq P2 + Y_ v(xi - xi) = 2M + NcMi + Hint ; i oo, reducing to just OPE. The results for the phase shifts in the lowest partial waves are presented in Fig. 1. Note the bad 1 So phase. To improve on this the long distance OPE transition potential is taken Vab;cd (R) = (tab • Tcd M Cab • CCD AB;CD(R) + [Si2]ab;cd [W|-]ab;cd (R)} , (18) where the tensor term is defined as S12 = 3(cab • R)(ctcd • R) — 4 4 = 4A © 20S © 20Ml © 20M2. Due to colour antisymmetry only the symmetric state survives which spin-isospin, (S,T), decomposition is20s = (7, 7) © (f, §) = N © A yielding N - A degeneracy. Since Ma — Mn is large at nuclear scales, one might still treat the Nucleon quartet N = (p T,p i,n T,n i) as the fundamental rep. of the old Wigner-Hund SU(4) symmetry which implies spin independence, in particular that Vi So (r) = VsSi (r) at all distances suggesting that phases 6i So (p) = 6sSl (p) in contradiction to data (see e.g. Fig. 1). The amazing finding of Ref. [6] was that assuming identical potentials Vi So (r) = VsSl (r) for r > rc —> 0 one has pcot6is (p =-c pcotSsg (p =-1 23 0 ai So C (p)+ D(p) 1 a3Si C (p) + D(p) where the functions A(p), B(p), C(p) and D(p) are identical in both channels, but the experimentally different scattering lengths aiSo = —23.74fm and a3Si = 3 Molecular methods used in the Skyrme model [36,37,12] are replaced by evaluating model form factor yielding regularized Meson Exchange potentials [38] where the only remnant of the model is in the meson-form factors. 5.42fm yield quite different phase shifts with a fairly good agreement. Thus, Wigner symmetry is broken by very short distance effects and hence corresponds to a long distance symmetry (a symmetry broken only by counterterms). Moreover, large Nc [23] suggests that Wigner symmetry holds only for even L, a fact verified by phase shift sum rules [6]. In Refs. [7,8] we analyze further the relation to the old Serber symmetry which follows from vanishing P-waves in S = 1 channels, showing how old nuclear symmetries are unveiled by coarse graining the NN interaction via the Vlowk framework [42] and with testable implications for Skyrme forces in mean field calculations [43]. The chiral quark model is supposedly an approximate non-perturbative description, but perturbative gluons may be introduced by standard minimal coupling [13], id —^ id + gAQ • AQ/2 with AQ the N? - 1 Gell-Mann colour matrices. A source of SU(4) breaking is the contact one gluon exchange which yields spin-colour chromo-magnetic interactions (Stj is the tensor operator), i(rr)__-_Sf ij 4mimjrij ij (24) breaking the A — N degeneracy. This short distance terms break also the 1S0 and 3Si degeneracy of the NN system providing an understanding of the long distance character of Wigner symmetry. Taking the Wigner symmetric zero energy state and perturbing around it, the previous argument suggests that 1 /as S1 — 1 /ai So = 0(Ma — Mn ) with a computable coefficient. 8 Conclusions Chiral Quark and Soliton models while quite different in appearance provide some universal behaviour regarding NN interactions. If the asymptotic potentials coincide, the main differences in describing the scattering data are due to a few low energy constants which in some cases are subjected to extreme fine tuning of the model parameters. The success of the model at finite energy is mainly reduced to reproducing these low energy parameters. One of us (E.R.A.) warmly thanks M. Rosina, B. Golli and S. ¡Sirca for the invitation and D. R. Entem, F. Fernandez, M. Pavon Valderrama and J. L. Goity for discussions. This work is supported by the Spanish DGI and FEDER funds with grant FIS2008-01143/FIS, Junta de Andalucía grant FQM225-05, and EU Integrated Infrastructure Initiative Hadron Physics Project contract RII3-CT-2004-506078. References 1. R. Machleidt, K. Holinde and C. Elster, Phys. Rept. 149 (1987) 1. 2. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D17 (1978) 768. 3. M. Pavon Valderrama and E. Ruiz Arriola, Phys. Rev. C72 (2005) 054002. 4. E. Ruiz Arriola, A. Calle Cordon and M. Pavon Valderrama, (2007), 0710.2770. 5. A. Calle Cordon and E. Ruiz Arriola, AIP Conf. Proc. 1030 (2008) 334. 6. A. Calle Cordon and E. Ruiz Arriola, Phys. Rev. C78 (2008) 054002. 7. A. Calle Cordon and E. Ruiz Arriola, Phys. Rev. C80 (2009) 014002. 8. E. Ruiz Arriola and A. Calle Cordon, (2009), 0904.4132. 9. A. Calle Cordon and E. Ruiz Arriola, (2009), 0905.4933. 10. M. Oka and K. Yazaki, Int. Rev. Nucl. Phys. 1 (1984) 489. 11. R.F. Alvarez-Estrada, F. Fernandez, J. L. Sanchez-Gomez and V. Vento, Lect. Notes Phys. 259 (1986) 1. 12. T.S. Walhout and J. Wambach, Int. J. Mod. Phys. E1 (1992) 665. 13. A. Valcarce, H. Garzilazo, F. Fernandez and P. Gonzalez, Rept. Prog. Phys. 68 (2005) 965. 14. N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001. 15. S. Aoki, T. Hatsuda and N. Ishii, (2009), 0909.5585. 16. H.B. O'Connell et al., Prog. Part. Nucl. Phys. 39 (1997) 201. 17. V.G.J. Stoks et al., Phys. Rev. C49 (1994) 2950. 18. M.B. Gavela et al., Phys. Lett. B82 (1979) 431. 19. O.W. Greenberg and H.J. Lipkin, Nucl. Phys. A370 (1981) 349. 20. C.V. Christov et al., Prog. Part. Nucl. Phys. 37 (1996) 91. 21. H. Weigel, Lect. Notes Phys. 743 (2008) 1. 22. E. Ruiz Arriola, W. Broniowski and B. Golli, Phys. Rev. D76 (2007) 014008. 23. D.B. Kaplan and A.V. Manohar, Phys. Rev. C56 (1997) 76. 24. M.C. Birse and M.K. Banerjee, Phys. Lett. B136 (1984) 284. 25. A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189. 26. L.Y. Glozman and D.O. Riska, Phys. Rept. 268 (1996) 263. 27. J.L. Goity, Phys. Atom. Nucl. 68 (2005) 624. 28. D. Bartz and F. Stancu, Phys. Rev. C63 (2001) 034001. 29. D. R. Entem, E. Ruiz Arriola, M. Pavon Valderrama and R. Machleidt, Phys. Rev. C77 (2008) 044006. 30. M.T. Fernandez-Carames, P. Gonzalez and A. Valcarce, Phys. Rev. C77 (2008) 054003. 31. G. Karl and J.E. Paton, Phys. Rev. D30 (1984) 238. 32. M. Pavon Valderrama and E. Ruiz Arriola, Phys. Rev. C74 (2006) 054001. 33. A.M. Green, Rept. Prog. Phys. 39 (1976) 1109. 34. G.H. Niephaus, M. Gari and B. Sommer, Phys. Rev. C20 (1979) 1096. 35. R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29 (1984) 1207. 36. N.R. Walet, R.D. Amado and A. Hosaka, Phys. Rev. Lett. 68 (1992) 3849. 37. N.R. Walet and R.D. Amado, Phys. Rev. C47 (1993) 498. 38. G. Holzwarth and R. Machleidt, Phys. Rev. C55 (1997) 1088. 39. N. Kaiser, S. Gerstendorfer and W. Weise, Nucl. Phys. A637 (1998) 395. 40. M. Pavon Valderrama and E. Ruiz Arriola, Annals Phys. 323 (2008) 1037. 41. M. Pavon Valderrama and E. Ruiz Arriola, Phys. Rev. C79 (2009) 044001. 42. S. K. Bogner, T. T. S. Kuo and A. Schwenk, Phys. Rept. 386 (2003) 1 43. M. Zalewski, J. Dobaczewski, W. Satula and T. R. Werner, Phys. Rev. C77 (2008) 024316 Bled Workshops in Physics Vol. 10, No. 1 p. 17 Electro-magnetic meson form-factor from a relativistic coupled-channels approach* E. P. Biernata, W. Schweigerb, K. Fuchsbergerb, and W. H. Klinkc a Institut für Physik, Universität Graz, A-8010 Graz, Austria b BE-OP Division, CERN, CH-1211 Geneve 23, Switzerland c Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA We calculate the electromagnetic form factor of a confined quark-antiquark pair within the framework of relativistic point-form quantum mechanics. The idea is to treat elastic electromagnetic scattering of an electron by a meson as a relativistic two-channels problem for a Bakamjian-Thomas type mass operator [1] such that the dynamics of the exchanged photon is taken explicitly into account. On the hadronic level the structure of the meson is encoded in a phenomeno-logical form factor which is not known a priori. Similarly, on the constituents level we can consider electromagnetic scattering of an electron by a confined quark-antiquark pair as a two-channels problem. The quark and the antiquark are assumed to interact via a spontaneous confining potential. Elimination of the channel containing the photon gives in both cases an eigenvalue equation for the eM and eqq channels on the hadronic and constituent levels, respectively, which contains the one-photon-exchange optical potential. In order to work within the Bakamjian-Thomas framework one has to resort to the approximation that the total four-velocity of the system is conserved at electromagnetic vertices [2]. By comparison of matrix elements of the optical potential on the hadronic and the constituent levels the electromagnetic meson form factor can be read off [3,4]. The form factor obtained in this way depends on all Lorentz invariants of the electron-meson system, i.e. on the momentum-transfer and on the total invariant mass of the electron-meson system. The dependence on the invariant mass is related to the violation of cluster separability. If, however, the invariant mass is chosen large enough this dependence becomes negligible. In the limit of an infinitely large invariant mass the optical potential separates into an electron and a meson current which are connected via the usual photon propagator. The expression for the form factor becomes then [5] F(Q2) = d%. ^SW*(k'q)W(kq). (1) J M/ m' Here Q2 = q2 is the momentum transfer squared with q = kq — kq = kM — kM and m^q = (Eq + Eq )2 — kM is the invariant mass of the quark-antiquark pair. * Talk delivered by E. P. Biernat 18 E. P. Biernat, W. Schweiger, K. Fuchsberger, and W. H. Klink Quantities without a tilde refer to the electron-meson center-of-mass and quantities with a tilde refer to the meson rest system. S is a spin-rotation factor which takes into account the substantial effect of the quark spin on the form factor. By an appropriate change of variables the integral for the form factor Eq. (1) takes the same form as the integral for the pion form factor from front form calculations [6,7]. This remarkable result means that relativity is treated in an equivalent way and the physical ingredients are the same in both approaches. For a simple two-parameter harmonic-oscillator wave function with the parameterization taken from [6,7] our result for the pion electromagnetic form factor provides a reasonable fit to the data as shown in Fig. 1. The generalization of this multichannel approach to electroweak form factors for an arbitrary bound few-body system is quite obvious. By an appropriate extension of the Hilbert space this approach is also able to accommodate exchange-current effects. Q2[GeV2] Q2[GeV2] Fig. 1. Q2 -dependence of the pion form factor with (solid) and without (dashed) spinrotation factor S. Values for the quark mass mq and the oscillator parameter a are taken from [6,7] and data are taken from [8-13]. References 1. B. Bakamjian and L. H. Thomas, Phys. Rev. 92,1300 (1953). 2. W. H. Klink, Nucl. Phys. A716,123 (2003). 3. K. Fuchsberger, Master's thesis, Karl-Franzens-Universitat Graz (2007). 4. E. P. Biernat, K. Fuchsberger, W. Schweiger, and W. H. Klink, Few Body Syst. 44, 311 (2008). 5. E .P. Biernat, W. Schweiger, K. Fuchsberger, and W. H. Klink, Phys. Rev. C79, 055203 (2009). 6. P. L. Chung, F. Coester, and W. N. Polyzou, Phys. Lett. B205, 545 (1988). 7. F. Coester, W. N. Polyzou, Phys. Rev. C71, 028202 (2005). 8. S. R. Amendolia et al, Nucl. Phys. B277,168 (1986) 9. C. N. Brown et al, Phys. Rev. D8, 92 (1973) 10. C. J. Bebek et al, Phys. Rev. D9,1229 (1974) 11. C. J. Bebek et al., Phys. Rev. D13, 25 (1976) 12. C. J. Bebek et al, Phys. Rev. D17,1693 (1978) 13. G. M. Huber et al, Phys. Rev. C78, 045203 (2008) Bled Workshops in Physics Vol. 10, No. 1 p. 20 Gravitational, electromagnetic, and transition form factors of the pion* Wojciech Broniowskia and Enrique Ruiz Arriolab a The H. Niewodniczanski Institute of Nuclear Physics PAN, PL-31342 Krakow and Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland b Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain Abstract. Results of the Spectral Quark Model for the gravitational, electromagnetic, and transition form factors of the pion are discussed. In this model both the parton distribution amplitude and the parton distribution function are flat, in agreement with the transverse lattice calculations at low renormalization scales. The model predictions for the gravitational form factor are compared to the lattice data, with good agreement. We also find a remarkable relation between the three form factors, holding within our model, which besides reproducing the anomaly, provides a relation between radii which is reasonably well fulfilled. Comparison with the CELLO, CLEO, and BaBar data for the transition form factor is also considered. While asymptotically the model goes above the perturbative QCD limit, in qualitative agreement with the BaBar data, it fails to accurately reproduce the data at intermediate momenta. The low-energy behavior of the pion is determined by the spontaneous breakdown of the chiral symmetry. This fact allows for modeling the soft matrix elements in a genuinely dynamical way [1-25]. This talk is based on Refs. [26,27] and employs the Spectral Quark Model (SQM) [28] in the analysis of several high-energy processes and their partonic interpretation. This model satisfies a priori consistency conditions [28] between open quark lines and closed quark lines, which becomes crucial in the analysis of high-energy processes and enables an unambiguous identification of parton distribution functions and amplitudes. This is not necessarily the feature of other versions of chiral quark models, such as the Nambu-Jona-Lasinio (NJL) model, as was spelled out already in Ref. [1]. For these reasons SQM is particularly well suited for the presented study. The general theoretical framework is set by the Generalized Parton Distributions (GPDs) [29-37]. These objects arise formally, e.g., from deeply virtual Comp-ton scattering (DVCS) on a hadronic target, effectively opening up the quark lines joining the currents. In local quark models usually the one-loop divergences appear and a regularization is needed. One may either compute the regularized DVCS and take the high-energy limit, or compute directly the regularized GPD. * Talk delivered by W. Broniowski Besides the requirements of gauge invariance and energy-momentum conservation, this apparently innocuous issue sets a non-trivial consistency condition on admissible regularizations which SQM fulfills satisfactorily. For the case of the pion, the GPD for the non-singlet channel is defined as £3ab Hq>NS (x,i,t): rf|_eixP+,-(7tb (p')|^(0)y+^(z)T3|7ra(p))|z+=Oizl with similar expressions for the singlet quarks and gluons. We omit the gauge link operators [0, z], absent in the light-cone gauge. The kinematics is set by p' = p + q, p2 = p '2 = m^, q2 = —2p • q = t. The variable Z = q+/p+ denotes the momentum fraction transferred along the light cone. Formal properties of GPDs can be elegantly written in the symmetric notation involving the variables f - _L- Y - ^ ~~ 2-C ~ 1-C/2 ' HJ=0(X, £,t) = —HJ=0(—X,£,t), HI=1 (X, £,t) = HI=1 (— X, £,t). For X > 0 one has HI=0>1 (X,0,0) = qS>NS(X), where q(x)i are the standard parton distribution functions (PDFs). In QCD all these objects are subjected to radiative corrections, as they carry anomalous dimensions, and become scale-dependent, i.e. they undergo a suitable QCD evolution. This raises an important question: what is the scale Q0 of the quark model when matching to QCD is performed? The momentum-fraction sum rule fixes this scale to be admittedly very low, Q0 = 313—20 MeV, for Aqcd = 226 MeV. Remarkably, but also perhaps unexpectedly, this choice, followed by the leading-order evolution, provides a rather impressive agreement with the high energy data, as well as the Euclidean and transverse-lattice simulations (see Ref. [26] for a detailed summary). The following sum rules hold for the moments of the GPDs: dX HI=1 (X, £, t) = 2Fv(t), dXX H (X, £, t) = 202 (t) — 2£201 (t), where FV (t) denotes the vector form factor, while 01(t) and 02(t) stand for the gravitational form factors [38]. Other important features are the polynomiality conditions [29], the positivity bounds [39,40], and a low-energy theorem [41]. We stress that all these properties required on formal grounds are satisfied in our quarkmodel calculation [26]. Unlike GPDs, the form factors of conserved currents do not undergo the QCD evolution. In the chiral limit we have the following identity in SQM relating the gravitational and electromagnetic form factor, A[tei(t)]=Fv(t), (1=1,2), (1) from which the identity between the two gravitational form factors 01 (t) = 02 (t) = 0(t) follows. Since there is no data for the full kinematic range for the GPDs of the pion, we present here the results for the generalized form factors only, in particular for 0 1 -t [GeV2] -t [GeV2] Fig. 1. Form factors of the pion vs. lattice data. Left: the electromagnetic form factor. Right: the quark part of the gravitational form factor, 0i (t)/2, computed in the Spectral Quark Model and compared to the lattice data from Ref. [42]. The band around the model curves indicates the uncertainty in the model parameters. the gravitational ones. It is well known that the data for the electromagnetic form factor are well parameterized with the monopole form, which by construction is reproduced in SQM, where the vector meson dominance is built in. The gravitational form factors are available from the lattice QCD simulations [42,43]. In Fig. 1 the electromagnetic form factor and the quark part of the gravitational form factor are compared to the lattice data. We note a very good agreement. In SQM one has the relation mp = 24n2f2/Nc, (2) where f is the pion weak decay constant in the chiral limit. This relation works within a few percent phenomenologically. The expressions for the form factors in SQM are very simple, mp mp I mp \ mp-1 t Vmp- V We note the longer tail of the gravitational form factor in the momentum space, meaning a more compact distribution of energy-momentum in the coordinate space. Explicitly, we find a quark-model formula 2(r2)e = (r2) v . (4) The two previous processes regard two pions and either one photon or one graviton in the corresponding three-point vertex function. An apparently disparate object is given by the pion-photon transition distribution amplitude (TDA) [44,45] dz- , , T N ,, tq . , N b i____t i e e-^Xp'.ejlrb^-^fzjlTt^p)) =+=0 = e^«P£vpaqpVab(x, i,t), 2 ZT =o p+f Here the photon carries momentum p' = p + q and has polarization e. As before, the presence of the gauge link operators is understood in Eq. (5) to guarantee the gauge invariance of the bilocal operators. We consider here the isovector quark bilinears. Since the photon couples to the quark through a combination of the isoscalar and isovector couplings, i.e. the quark charge is Q = 1 /(2Nc) + t3/2, one has the isospin decomposition V ab(x,i,t) = 6abVi=c (x,Z,t)+ ieabcVi=1 (x,Z,t). (6) The isoscalar form factor is related to the pion-photon transition form factor by the sum rule l~7tYY* (t) — — dxV I=0(x,C,t), (7) where the factor of 2 comes fom the fact, that either of the photons can be isoscalar. The form factor in SQM was obtained directly in Ref. [28] and later on from the integration of the pion-photon isoscalar transition distribution amplitude (TDA) yielding [21] a —independent function (as required by polynomiality), 2f 1~7tyy (t) A) = —— N 2m2p 1 / 2m2 — (1 -A)t m4 — tm2 + (1 — A2)t2 At °S 1 2m2 - (1 +A)t , (8) where A = (q? — q2)/(q? + q2) is the photon asymmetry parameter. For A we have F^yy* (t) 12n2f m2 — t t Up (9) where relation (2) has been used. We read out from this formula the corresponding rms radius to be (r2)^y2y, = v^/nip = 0.57 fm for mp = 770 MeV. Equiva- SQM lently, one may use the slope parameter b„ = ^F^Oyy* (tj/F^Oyy* (t) dt nuYY* 1 nuyy* gives bn = 5/(6mp) = 1.4 GeV-2, in a very reasonable agreement with the experimental value bn = (1.79 ± 0.14 ± 14)GeV-2, originally reported by CELLO [46]. A comparison of Eq. (8,9) to the CLEO [47] and BaBar [48] data is presented in the right panel of Fig. 2. The solid line corresponds to the model calculation with A = 1, while the dashed line is for A = 0.95. We note that the experiment does not produce strictly real photons, thus the observed sensitivity to the value of A is a relevant effect. We note that while at |A| = 1 the model asymptotics for the transition form factor is (2f/Nc) log(—t/mp)/(—t), at |A| = 1 it becomes (2f/Nc) log[(1 + A)/(1 — A)]/(—At). The behavior is clearly seen in Fig. 2. As we notice, in the intermediate range of Q SQM overshoots the data. The recent BaBar measurements [48] have predated the long-standing per-turbative QCD prediction [49,50] that —tFnYY* (t) goes asymptotically to a constant value of 2f. Some authors [51,52] have pointed out that the key to this unexpected behavior hints for a flat pion PDA and the end-point singularities and switched-off QCD evolution. The flatness of the PDA at low renormalization scales has been originally found in the Nambu-Jona-Lasinio model [10] and in SQM [28]. t=0 c X Q2 [GeV] Fig. 2. Left: chiral quark model prediction for the pion DA evolved to the scale of 0.5GeV (band) and compared to the transverse lattice data [54]. Right: the pion trantition form factor compared to the CLEO [47] and BaBar [48] data. Solid (dashed) lines are the SQM prediction at A = 1 (A = 0.95). The dotted line is the perturbative QCD prediction. We note in passing that a constant PDA is also found in the Regge model [53]. Remarkably, an almost flat PDA is also found non-perturbatively on the transverse lattice [54] (see the left panel of Fig. 2). Actually, the non-vanishing of the PDA at the end points (at the quark-model scale) is not only a consequence of local quark models. Nonlocal models correctly implementing the chiral Ward-Takahashi identity also get such a feature [18]. A trend to flatness is observed in contrast to calculations violating the chiral symmetry constraints. However, the corresponding transition form factor in non-local models does not show a steep rise [55] as suggested by the BaBar data. The calculation in Ref. [56,57], which reproduces the CLEO and BaBar data, requires, unfortunately, a much too small constituent quark mass, which is incompatible with other sectors of the pion phenomenology. The apparent inconsistency of the BaBar data with the QCD convolution scheme is also addressed in Ref. [58,59]. Let us remind the reader that according to the conventional perturbative QCD approach, the radiative corrections are computed order by order in the twist expansion. Most often this is in practice possible only for the leading-twist contribution. Actually, this is the only way to identify the PDA within a non-perturbative scenario or quark model calculations. In fact, the chiral quark models require a low scale not only by fixing the second Gegenbauer coefficient a2 of the PDA. As already mentioned, the same conclusion is reached independently by fixing the momentum fraction of the valence quarks to its natural 100% value at the quark-model scale, where the quarks constitute the only degrees of freedom. On a more methodological level, it is worth mentioning that the conventional NJL model does not share some of the virtues of SQM, particularly the interplay between chiral anomaly and factorization, a subtle point which was discussed at length in Ref. [11] for the NJL case. The nyy triangle graph is linearly divergent, and thus a regularization must generally be introduced. If one insists on preserving the vector gauge invariance, the regulator must preserve that symmetry, but then the axial current is not conserved, generating the standard chiral anomaly. The obvious question arises whether the limit Q2 —> oo must be taken before or after removing the cut-off. If one takes the sequence Q2 > A2, a constant PDA is obtained in agreement with our low energy calculation. For the opposite sequence factorization does not hold in NJL. The good feature of SQM is that the spectral regularization does not make any difference between the two ways. This illustrates in a particular case the above-mentioned general consistency requirement between regularized open and closed quark lines (see e.g. [60]). Finally, by combining Eq. (3) and Eq. (9) we get the remarkable relation among the electromagnetic, gravitational and transition form factors, holding in SQM: p7tYY*(t) = T¿? [2Fv(t) + 0(t)] ' (10) whence 3(r2)„YY* = 2(r2)y + (r2)@ . (11) The previous relation is not fulfilled in the conventional NJL model. Of course, it would be interesting to test the relation Eq. (10) against the future data or lattice QCD. In conclusion, we note that while the description of the pion transition form factor in a genuinely dynamical way remains a challenge, the Spectral Quark Model offers many attractive features which are required from theoretical consistency. It satisfies the chiral anomaly and the factorization property. The vector and gravitational form factors describe experimental and/or lattice-QCD data satisfactorily. A remarkable model relation among the gravitational, electromagnetic and transition form factors has also been deduced. Finally, for the latter, we have also displayed a hitherto unnoticed sensitivity to the photon momentum asymmetry parameter A which might be relevant for other studies. This research is supported in part by the Polish Ministry of Science and Higher Education, grants N202 034 32/0918 and N202 249235, Spanish DGI and FEDER funds with grant FIS2008-01143/FIS, Junta de Andalucía grant FQM225-05, and the EU Integrated Infrastructure Initiative Hadron Physics Project, contract RII3-CT-2004-506078. References 1. R.M. Davidson and E. Ruiz Arriola, Phys. Lett. B348 (1995) 163. 2. A.E. Dorokhov and L. Tomio, (1998), hep-ph/9803329. 3. M.V. Polyakov and C. Weiss, Phys. Rev. D59 (1999) 091502, hep-ph/9806390. 4. M.V. Polyakov and C. Weiss, Phys. Rev. D60 (1999) 114017, hep-ph/9902451. 5. A.E. Dorokhov and L. Tomio, Phys. Rev. D62 (2000) 014016. 6. I.V. Anikin et al., Nucl. Phys. A678 (2000) 175. 7. I.V. Anikin et al., Phys. Atom. Nucl. 63 (2000) 489. 8. E. Ruiz Arriola, (2001), hep-ph/0107087. 9. R.M. Davidson and E. Ruiz Arriola, Acta Phys. Polon. B33 (2002) 1791, hep-ph/0110291. 10. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D66 (2002) 094016, hep-ph/0207266. 11. E. Ruiz Arriola, Acta Phys. Polon. B33 (2002) 4443, hep-ph/0210007. 12. M. Praszalowicz and A. Rostworowski, (2002), hep-ph/0205177. 13. B.C. Tiburzi and G.A. Miller, Phys. Rev. D67 (2003) 013010, hep-ph/0209178. 14. B.C. Tiburzi and G.A. Miller, Phys. Rev. D67 (2003) 113004, hep-ph/0212238. 15. L. Theussl, S. Noguera and V. Vento, Eur. Phys. J. A20 (2004) 483, nucl-th/0211036. 16. W. Broniowski and E. Ruiz Arriola, Phys. Lett. B574 (2003) 57, hep-ph/0307198. 17. M. Praszalowicz and A. Rostworowski, Acta Phys. Polon. B34 (2003) 2699, hep-ph/0302269. 18. A. Bzdak and M. Praszalowicz, Acta Phys. Polon. B34 (2003) 3401, hep-ph/0305217. 19. S. Noguera and V. Vento, Eur. Phys. J. A28 (2006) 227, hep-ph/0505102. 20. B.C. Tiburzi, Phys. Rev. D72 (2005) 094001, hep-ph/0508112. 21. W. Broniowski and E.R. Arriola, Phys. Lett. B649 (2007) 49, hep-ph/0701243. 22. A. Courtoy and S. Noguera, Phys. Rev. D76 (2007) 094026, 0707.3366. 23. A. Courtoy and S. Noguera, Prog. Part. Nucl. Phys. 61 (2008) 170, 0803.3524. 24. A.E. Dorokhov and W. Broniowski, Phys. Rev. D78 (2008) 073011, 0805.0760. 25. P. Kotko and M. Praszalowicz, (2008), 0803.2847. 26. W. Broniowski, E.R. Arriola and K. Golec-Biernat, Phys. Rev. D77 (2008) 034023, 0712.1012. 27. W. Broniowski and E.R. Arriola, Phys. Rev. D78 (2008) 094011, 0809.1744. 28. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D67 (2003) 074021, hep-ph/0301202. 29. X.D. Ji, J. Phys. G24 (1998) 1181, hep-ph/9807358. 30. A.V. Radyushkin, (2000), hep-ph/0101225. 31. K. Goeke, M.V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47 (2001) 401, hep-ph/0106012. 32. A.P. Bakulev et al., Phys. Rev. D62 (2000) 054018, hep-ph/0004111. 33. M. Diehl, Phys. Rept. 388 (2003) 41, hep-ph/0307382. 34. X.D. Ji, Ann. Rev. Nucl. Part. Sci. 54 (2004) 413. 35. A.V. Belitsky and A.V. Radyushkin, Phys. Rept. 418 (2005) 1, hep-ph/0504030. 36. T. Feldmann, Eur. Phys. J. Special Topics 140 (2007) 135. 37. S. Boffi and B. Pasquini, Riv. Nuovo Cim. 30 (2007) 387, 0711.2625. 38. J.F. Donoghue and H. Leutwyler, Z. Phys. C52 (1991) 343. 39. B. Pire, J. Soffer and O. Teryaev, Eur. Phys. J. C8 (1999) 103, hep-ph/9804284. 40. P.V. Pobylitsa, Phys. Rev. D65 (2002) 077504, hep-ph/0112322. 41. M.V. Polyakov, Nucl. Phys. B555 (1999) 231, hep-ph/9809483. 42. D. Brommel, Pion structure from the lattice, PhD thesis, University of Regensburg, Regensburg, Germany, 2007, DESY-THESIS-2007-023. 43. D. Brommel et al., PoS LAT2005 (2006) 360, hep-lat/0509133. 44. J.P. Lansberg, B. Pire and L. Szymanowski, Phys. Rev. D73 (2006) 074014, hep-ph/0602195. 45. J.P. Lansberg, B. Pire and L. Szymanowski, (2007), 0709.2567. 46. CELLO, H.J. Behrend et al., Z. Phys. C49 (1991) 401. 47. CLEO, J. Gronberg et al., Phys. Rev. D57 (1998) 33, hep-ex/9707031. 48. The BABAR, B. Aubert et al., Phys. Rev. D80 (2009) 052002, 0905.4778. 49. A.V. Efremov and A.V. Radyushkin, Phys. Lett. B94 (1980) 245. 50. G.P. Lepage and S.J. Brodsky, Phys. Lett. B87 (1979) 359. 51. A.V. Radyushkin, (2009), 0906.0323. 52. M.V. Polyakov, (2009), 0906.0538. 53. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D74 (2006) 034008, hep-ph/0605318. 54. S. Dalley and B. van de Sande, Phys. Rev. D67 (2003) 114507, hep-ph/0212086. 55. P. Kotko and M. Praszalowicz, (2009), 0907.4044. 56. A.E. Dorokhov, (2009), 0905.4577. 57. A.E. Dorokhov, (2009), 0909.5111. 58. S.V. Mikhailov and N.G. Stefanis, Nucl. Phys. B821 (2009) 291, 0905.4004. 59. S.V. Mikhailov and N.G. Stefanis, (2009), 0909.5128. 60. E.R. Arriola, W. Broniowski and B. Golli, Phys. Rev. D76 (2007) 014008, hep-ph/0610289. Bled Workshops in Physics Vol. 10, No. 1 p. 28 Axial charges of nucleon resonances* Ki-Seok Choi, W. Plessas, and R.F. Wagenbrunn Theoretical Physics, Institute of Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria Recently, first results have become available from lattice quantum chromodynam-ics (QCD) for two of the nucleon excitations, namely, the negative-parity N * (1535) and N*(1650) resonances [1]. The axial charge of the nucleon ground state had been studied before by different lattice-QCD groups in quenched calculations and with dynamical quarks [2-7]. In some of these works one has used chiral extrapolations (for a recent discussion of the associated problems see Ref. [8]), and the bulk of results obtained for gA of the nucleon varies between about 1.10 - 1.40. Lately, the issue of axial constants of N* resonances has become debated a lot due to the suggestion of chiral-symmetry restoration in the higher hadron spectra [9,10]. According to this scenario there should appear chiral doublets of positive- and negative-parity states and as a further consequence their axial charges should became small or almost vanishing. The first parity partners above the nucleon ground state are supposed to be the N*(1440)-N*(1535), the next ones the N*(1710)-N*(1650). The axial charges of the negative-parity partners in these pairs have been calculated in lattice QCD to be -0.00 and -0.55, respectively [1]; for the positive-parity states no results are yet available. We have performed a study of the axial charges of N* resonances in the framework of the relativistic constituent quark model (RCQM). Specifically we have extended a previous investigation of the nucleon axial form factors [11,12] to the first Jp = ^ nucleon excitations. Our approach relies on solving the eigenvalue problem of the Poincare-invariant mass operator in the framework of rela-tivistic quantum mechanics. The axial current operator is chosen according to the spectator model (SM) [13]. For the RCQM we employed in the first instance the extended Goldstone-boson exchange (EGBE) RCQM [14], as it produces the most elaborate nucleon and N* wave functions. In Table 1 we present a selection of results for the axial charges gA of the nucleon and the N*(1440), N*(1710), N*(1535), as well as N*(1650) resonances in case of the EGBE RCQM. It is immediately evident that the EGBE RCQM produces reasonable values for the axial charges in all instances without any further fittings. In the cases where a comparison is possible it produces the same pattern as lattice QCD. The gA of the nucleon and of N*(1440) are practically of the same size, with the theoretical result for the nucleon being quite close to the experi- * Talk delivered by Ki-Seok Choi mental value of gA=1.2695±0.0029 [15]. The nonrelativistic calculations cannot produce this value, neither in the simplistic SU(6) x O(3) quark model nor in the nonrelativistic limit of the RCQM. For the negative-parity N * (1535) resonance the gA is predicted to be compatible with 0, while for the negative-parity N*(1650) resonance it is 0.51; both cases agree with the lattice-QCD results of Ref. [1]. Accidentally, the gA value of the nonrelativistic SU(6) x O(3) quark model is similar in the N*(1650) case but the nonrelativistic limit of the EGBE RCQM shows deviations for both of the \ resonances. At this time nothing is known from lattice QCD for the j+ resonances. For the latter, it would also be most interesting to check our results against lattice QCD, and we look forward to corresponding calculations. Table 1. Predictions for axial charges gA of the EGBE in comparison to available lattice QCD results [1-7], the values calculated by Glozman and Nefediev [9] within the SU(6) x O(3) nonrelativistic quark model, and the nonrelativistic limit from the EGBE RCQM. State JP EGBE Lattice QCD SU(6) x O(3) QM EGBE nonrel N(939) 1 + 1.15 1.10-1.40 1.66 1.65 N(1440) 1 + N(1535) 1.16 0.02 0.00 1.66 -0.11 1.61 -0.20 N(1710) 1 + N(1650) 0.35 0.51 0.55 0.33 0.55 0.42 0.64 It is particularly satisfying to find the RCQM predictions for the axial charges of the N*(1535) and N*(1650) resonances in agreement with the lattice-QCD results. We may thus be confident that at least for zero momentum-transfer processes the mass eigenstates of these nucleon excitations as produced especially with EGBE RCQM are quite reasonable. The latter is supposed to model the SBxS property of low-energy QCD. This type of hyperfine interaction, which also introduces an explicit flavor dependence, has been remarkably successful in describing a number of phenomena in low-energy baryon physics. Most prominently, it produces the correct level orderings of the positive- and negative-parity N* resonances and simultaneously the ones in the other hyperon spectra, notably the A spectrum. The RCQM with GBE dynamics does not have any mechanism for chiral-symmetry restoration built in. As such it cannot be expected to produce parity doublets due to this reason. Nevertheless the EGBE RCQM describes the N* resonance masses with good accuracy (mostly within the experimental error bars or at most exceeding them by 4%). Acknowledgments K-S. C. is grateful to the organizers of the Mini-Workshop for the invitation and for creating such a lively working atmosphere. This work was supported by the Austrian Science Fund (through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars; FWF DK W1203-N08). References 1. T. T. Takahashi and T. Kunihiro, Phys. Rev. D 78, 011503 (2008). 2. D. Dolgov et al. [LHPC collaboration and TXL Collaboration], Phys. Rev. D 66, 034506 (2002). 3. R. G. Edwards et al. [LHPC Collaboration], Phys. Rev. Lett. 96, 052001 (2006). 4. A. A. Khan et al, Phys. Rev. D 74, 094508 (2006). 5. T. Yamazaki et al. [RBC+UKQCD Collaboration], Phys. Rev. Lett. 100,171602 (2008). 6. H. W. Lin, T. Blum, S. Ohta, S. Sasaki and T. Yamazaki, Phys. Rev. D 78, 014505 (2008). 7. C. Alexandrou etal, PoS(LATTICE 2008), 139 (2008); arXiv:0811.0724 [hep-lat]. 8. V. Bernard, Prog. Part. Nucl. Phys. 60, 82 (2008). 9. L. Y. Glozman, Phys. Rept. 444, 1 (2007). 10. L. Y. Glozman and A. V. Nefediev, Nucl. Phys. A 807, 38 (2008). 11. L. Y. Glozman, M. Radici, R. F. Wagenbrunn, S. Boffi, W. Klink and W. Plessas, Phys. Lett. B 516,183 (2001). 12. S. Boffi, L. Y. Glozman, W. Klink, W. Plessas, M. Radici and R. F. Wagenbrunn, Eur. Phys. J. A 14,17(2002). 13. T. Melde, L. Canton, W. Plessas and R. F. Wagenbrunn, Eur. Phys. J. A 25, 97 (2005). 14. K. Glantschnig, R. Kainhofer, W. Plessas, B. Sengl and R. F. Wagenbrunn, Eur. Phys. J. A 23, 507 (2005). 15. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667,1 (2008). Bled Workshops in Physics Vol. 10, No. 1 p. 31 Nucleon axial couplings and [(1,0) e(0,i)H(i,i) 0(1,1)] chiral multiplet mixing* V. Dmitrasinovica, A. Hosakab, K. Nagatac a Vinca Institute of Nuclear Sciences, lab 010, P.O.Box 522,11001 Beograd, Serbia b Research Center for Nuclear Physics, Osaka University, Mihogaoka 10-1, Osaka 567-0047, Japan c Research Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, Japan Abstract. Three-quark nucleon interpolating fields in QCD have well-defined, UA ( 1 ) and SUl(2) x SUr(2) chiral transformation properties: mixing of the [(1, j) © (j, 1 )] chiral multiplet with one (of four available) [(1,0) © (0,1)], or [(0,1) © (j,0)] fields can be used to fit the isovector axial coupling gA' and thus predict the isoscalar axial coupling g A' of the nucleon, in reasonable agreement with experiment. We also use a chiral mesonbaryon interaction to calculate the masses and one-pion-interaction terms of J = j baryons belonging to the [(0,1) © (j, 0)] and [(1, j) © (j, 1)] chiral multiplets and fit two of the diagonalized masses to the lowest-lying nucleon resonances thus predicting the third J = j resonance at 2030 MeV, not far from the (one-star PDG) state A(2150). 1 Introduction Almost 40 years ago Weinberg [1] considered mixing of chiral multiplets in the broken symmetry phase. In general such representation mixing may be complicated, but if only a few states are mixed, it may have predictive power. For instance, Weinberg used the mixing of [(-j,0) © (0, -j)] and [(1, \ ) © 1)] to explain the nucleon's isovector axial coupling constant gA' = 1.23, its value at the time (the present value being 1.267). Weinberg's idea predated QCD and did not even invoke the existence of quarks, but it may still be viable in QCD. Indeed, this idea was revived in the early 1990's, since when it has been known by the name of mended symmetry [2]. The nucleon also has an isoscalar axial coupling gA', which has been estimated from spin-polarized lepton-nucleon DIS data as gA' = 0.28 ± 0.16 [3], or the more recent value 0.33 ± 0.03 ± 0.05 [4]. The question is if the same chiral mixing angle can also explain the anomalously low value of this coupling? The answer manifestly depends on the UA (1) chiral transformation properties of the two admixed nucleon fields. * Talk delivered by V. Dmitrasinovic In this paper we address this question using the UA (1) chiral transformation properties of nucleon fields [6,7] as derived from the three-quark nucleon interpolating fields in QCD. If the answer to our question turns out in the positive, we may speak about Weinberg's idea being viable in QCD. To test the present idea, besides the phenomenological study, we also investigate an extended linear sigma model containing baryon resonances, where we evaluate the axial couplings using baryon masses as input. 2 Three-quark nucleon interpolating fields We start by summarizing the transformation properties of various quark trilinear forms with quantum numbers of the nucleon as shown in Refs. [6,7]. It turns out that every nucleon, i.e., spin- and isospin 1/2 field, besides having same non-Abelian transformation properties, comes in two varieties: one with "mirror" and another with "triple-naive" Abelian chiral properties. In Table 1 we show the Abelian and non-Abelian chiral properties of the nucleon interpolating fields in QCD, Ref. [6,7]. Here we shall use those results as the theoretical input into our calculations. This constitutes a minimal assumption, as one has no other guide to the chiral representations of the nucleon. In Refs. [57] the local (non-derivative) spin j baryon operators Nl = 6abc(qaqb)qc, (1) N2 = eabc(q aY5qb )y5qc, (2) were classified according to their Lorentz, chiral SUL (2) x SUR (2) (so-called "naive" chiral multiplet, whose axial charge is positive) and UA (1) group representations. Here we have introduced the "tilde-transposed" quark field q as q = qT Cy5(iT2), where C = iy2y0 is the Dirac field charge conjugation operator, t2 is the second isospin Pauli matrix. Once one allows for the presence of one derivative, such as the so-called "mirror" (0, j) © [\,0), whose axial charge is negative, Ref.[8], 2 2 N? = eabc(q a qb)i3Hy^qc, (3) N2 = £abc(q a y5qb )i9HyHy5qc. (4) and the (1, -j) ffi (-j, 1) nucleon chiral representation N3 = ia-qyvq^ysq, (5) N4 = ia-qyvys^q)^ Tq (6) also become Pauli allowed, see Table 1. Here = — We found that indeed, as Gell-Mann and Levy [9] had postulated, the lowest-twist (nonderivative) }= j nucleon field(s) form a (^,0) chiral multiplet, albeit there are two such independent fields. There is only one set of }= \ Pauli-allowed sub-leading-twist (one-derivative) interpolating fields that form a (1, j) chiral multiplet, however. Table 1. The Abelian and the non-Abelian axial charges (+ sign indicates "naive", sign "mirror" transformation properties) and the non-Abelian chiral mutiplets of JP = j, Lorentz representation (j,0) nucléon fields. The field denoted by 0 belongs to the ( 1, j ) © ( j, 1 ) chiral multiplet and is the basic nucléon field that is mixed with various ( j, 0) nucléon fields in Eq. (7). case field (0) 9A y a SUl(2) x SUr(2) I Ni - N2 -1 +1 (^,0)©(0,1) II Ni + N 2 +3 +1 (^,0)©(0,1) III Ni - N2 +1 -1 (0, !)©(! 0) IV Nh +N2 -3 -i (0, !)©(! 0) 0 N; + IN; +1 +t (1, })©(},U 3 Mixing of two chiral representations Next consider the mixing of one of the fundamental chiral representations, as shown in Table 1 and the "higher" representation (1, j) for the nucleon, 9a 'mix. = 9a,'a cos20+ 9^,1,4) ^ 0> = g^1 a cos2 9 + | sin2 9 = 1.267. (7) Here the suffix a corresponds to one of I-IV and the corresponding values of g A' a are given in Table 1. We have also used the fact that g^1 ± ( = §, see Ref. [1,7]. This provides a possible solution to the nucleon's axial coupling problem in QCD. Three-quark nucleon interpolating fields in QCD have well-defined twofold UA(1) chiral transformation properties, see Table 1, that can be used to (naively) predict the isoscalar axial coupling gA!nix as follows gAL. = gk01« cos2e + gA0)(li,, sin2e, (8) together with the mixing angle 0 extracted from Eq. (7). Note, however, that due to the different (bare) non-Abelian gA' and Abelian gA' axial couplings, see Table 1, the mixing formulae Eq. (8) give substantially different predictions from one case to another, see Table 2. We can see in Table 2 that the two candidates are cases I and IV, with gA' = -0.2 and gA' = 0.4, respectively, the latter being within 1-ct of the measured value gA' = 0.33 ± 0.08. The nucleon field in case I is the well-known "Ioffe current", which reproduces the nucleon's properties in QCD lattice and sum rules calculations. The nucleon field in case IV is a "mirror" opposite of the orthogonal complement to the Ioffe current, an interpolating field that, to our knowledge, has not been used in QCD thus far. 3.1 A Simple Model The next step is to try and reproduce this phenomenological mixing starting from a model interaction, rather than per fiat. As the first step in that direction we Table 2. The values of the baryon isoscalar axial coupling constant predicted from the naive mixing and gA 'expt = 1.267; compare with gA>iXpt = 0-33 ± 0.03 ± 0.05. case (9A\9A0J) 9 (1) 0 A mix. ° (0) 9 A mix. (0) 9 A mix. I (+1,-1) 4(4- -cos20) ±39.3° - cos 20 -0.20 II (+1, +3) - cos 20) ±39.3o 2 + cos 20 2.20 III (-1, +1) -4 cos 20) ±67.2o 1 1.00 IV (-1,-3) -4 cos 20) ±67.2° -(1 + 2 cos 20) 0.40 must look for a dynamical source of mixing. One such mechanism is the simplest chirally symmetric non-derivative one-(a, n)-meson interaction Lagrangian, which induces baryon masses via its a-meson coupling. We shall show that only the mirror fields couple to the (1, j) baryon chiral multiplet by non-derivative terms; the naive ones require one (or odd number of) derivative. This is interesting, as we have already pointed out that the mixing case IV seems a preferable one from the phenomenological consideration of axial couplings. We use the projection method of Ref. [10] to construct the chirally invariant diagonal and off-diagonal meson-baryon-baryon interactions involving the "mirror" baryon Bi e (0,1), the (B2, A) e (1, \ ) baryon and one (ct, 7t) e ij,j) meson chiral multiplets. Here all baryons have spin 1/2, while the isospin of B1 and B2 is 1/2 and that of A is 3/2. The A field is then represented by an isovector-isospinor field A1, (i = 1,2,3). We found that for non-derivative mixing interaction the following chirally invariant combination L3 ga Bi (0- + -y5T • tt)B2 + 4Bt iysTtW + h.c. (9) with the coupling constant g3 induces an off-diagonal term in the baryon mass matrix after spontaneous symmetry breaking (a}0 —> fn via its a-meson coupling. Of course this is in addition to the conventional diagonal interactions: Li L2 giB 1 (a - iY5T • n) Bi, 2 ~392 5 • B2(CT + -lAMa + iys-r-TTjA1 ^FB2T1(c + iy5T- ^A1 + h.c. 3 (10) (11) In writing down the Lagrangians (9,10,11), we have implicitly assumed that the parities of B1, B2 and A are the same. In principle, their parities are arbitrary, except for the parity of the ground state nucleon, which must be even. For instance, if B2 has odd parity, the first term in the interaction Lagrangian Eq. (9) must include another y5 matrix. Here we consider all possible cases for the parities of B2 and A. Having established the mixing interaction Eq. (9), as well as the diagonal terms Eqs. (10),(11), we calculate the masses of the baryon states, as functions of the pion decay constant/chiral order parameter and (as yet undetermined) Born approximation coupling constants. We diagonalize the mass matrix and express the mixing angle in terms of diagonalized masses. We find the following double-angle formulas for the mixing angles 0i,...,4, in the four different parities scenarios tan 20! = \J (2N + A) (2N*~ — A) (A-N+N*-) (12) tan 202 = V/(A-2N)(2N*+ - A) (13) (N +N*+ - A) tan 203 = — (2N - A)(2N*- + A) (14) (A-N+N*-) tan 204 = vMA + 2N)(2N*+ + A) (N + N*+ + A) (15) where N is the nucleon ground state mass (940 MeV) and N*±, A are the masses of the nucleon excited state, where ± indicates the parity of the N* state. These angles correspond to the two (variable) parities as follows 0? <-> (N*-, A+), 02 <-> (N*+, A-), 03 ^ (N*-, A-), 04 ^ (N*+ ,A+), where ± indicates the parity of the state. Note that the angle 04 is necessarily imaginary so long as the A,N* masses are physical (positive), and that the reality of the mixing angle(s) imposes stringent limits on the A, N* resonance masses in other three cases, as well. Next, we use (some of) the observed resonance masses to determine the mixing angle(s) and thence calculate the axial couplings. 3.2 Results Direct prediction The four lowest-lying (besides the N(940)) candidate states in the PDG tables are: R(1440), N(1535), A(1620), A(1910), we use them to fit the free coupling constants. Of the two "mass allowed" scenarios, however, none survive the axial coupling test. Perhaps our choice of input resonances is inadequate. Note that one may "invert" this procedure, however, and use the isovector axial coupling to predict one of the baryon masses, say the A's, having fixed the other two, in this case the nucleon's N(940) and N*(1440) or N*(1535). Inverse prediction Next, we use the double-angle formulas Eqs. (12)-(15) for the mixing angles 02,...,4 together with the two observed nucleon masses to predict the A masses shown in the Table 3. We see that only the (N*+, A-) parity case leads to a realistic prediction: The difference between the observed (one-star) S31 (2150) [11] A resonance mass and the predicted 2030 MeV may be neglected in view of the uncertainties and typical widths of states at such (high) energies. We shall not attach undue significance to this proximity in view of the rather uncertain status of this resonance, at least not until it is confirmed by another experiment. This choice of resonances leads to a reasonable nNN coupling constant (14.2 vs. 13.6 expt.) and predicts a set of as yet not measured n-baryon couplings. Table 3. The values of the A baryon masses predicted from the isovector axial coupling gA L. = gA 'expt. = 1.267 and g AL. = 0.4 vs. g ALp, = 0.33 ± 0.08. (N*p, Ap') (N, N*) A(MeV) expt~ (-,+) N(940), R(1535) 2330 1910 (+, -) N(940), R(1440) 2030,2730 1620,2150 (-,-) N(940), R(1535) 1140 1620,2150 A comment about the comparatively high value of the A mass seems to be in order now: In the mid-1960-s Hara [12] noticed that the chiral transformation rules for a (1, j) mulfiplef impose a strict and seemingly improbable mass relation among its two members: m^ = 2m,N. The mixing with the (j, 0) multiplet modifies this mass relation for the worse, i.e. it makes the A even heavier. For this reason, the lowest-lying A's of either parity cannot be the chiral partners of the nucleon ground state, as we initially assumed in our "direct prediction". 4 Three-field mixing A linear superposition of yet another field (except for the mixture of cases II and III above) ought to give a perfect fit to both experimental values. Such an admixture introduces new free parameters (besides the two already introduced mixing angles, e.g. 0i and 04, we have the relative/mutual mixing angle 014, as the two nucleon fields I and IV may also mix). One may subsume/redefine the sum and the difference of the two angles 0i and 04 into the new angle 0, whereas one may define 0i4 = 9 (this relationship depends on the precise definition of the mixing angles 0i, 04 and 0i4); thus we find two equations with two unknowns of the general form: - sin20 + cos20 (gA 'cos2(p + gA 'Wcp) = 1.267 (16) sin20 + cos20 (gA°'cos2 9 + gA''sin2= 0.33 (17) The values of the mixing angles obtained from this simple fit to the two baryon axial coupling constants are shown in Table 4. This, however, is not just a mere fit: when extending to the SUL (3) x SUR (3) symmetry, chiral transformation properties of the nucleon fields differ: Ni -N2 e [(3,3)® (3,3)], Ni + N2 e [(8,1)® (1,8)] and (N3 + jN^) e [(6,3) © (3,6)], see Ref. [13]. From these chiral SUL (3) x SUR (3) symmetry assignment we can also predict the F and D couplings (un-corrected for the explicit SUF(3) symmetry breaking) in Table 4, which can be compared with the experimental numbers. We have not calculated the SUF(3) symmetry breaking corrections, as yet, so we have not taken into account the "error bars" on the mixing angle(s), which remains a task for the future. At any rate, it should be clear that the predicted values are "in the right ball park" for most of the scenarios considered here. Thus, the chiral multiplet mixing remains a viable theoretical scenario for the explanation of the nucleon isoscalar axial couplings. Table 4. The values of the mixing angles obtained from the fit to the baryon axial coupling constants and the predicted values of axial F and D couplings. Experimental values have evolved from F=0.459 ± 0.008 and D=0.798 ± 0.008 in Ref. [14] to F=0.477 ± 0.001 and D=0.835 ± 0.001 in Ref. [15]. Note that the new values are more than 2-ff away from the old ones, and that the new F,D add up to F+D = 1.312 = 1.269 ± 0.002. case (0) 9 a expt. q(!) 9 A expt. e