Blejske delavnice iz fizike Bled Workshops in Physics ISSN 1580-4992 Letnik 2, St. 1 VOL. 2, NO. 1 Proceedings of the Mini-Workshop Selected Few-Body Problems in Hadronic and Atomic Physics Bled, Slovenia, July 7-14, 2001 Edited by Bojan Golli1'2 Rajmund Krivec2 Mitja Rosina1'2 1 University of Ljubljana, 2J. Stefan Institute DMFA - ZALOŽNIŠTVO Ljubljana, december 2001 The Mini-Workshop Selected Few-Body Problems in Hadronic and Atomic Physics was organized by Joief Stefan Institute, Ljubljana Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana and sponsored by Ministry of Education, Science and Sport of Slovenia Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Organizing Committee Rajmund Krivec Mitja Rosina Bojan Golli List of participants Nir Barnea, Jerusalem, nir@vms . huj i.ac.il Wojciech Broniowski, Cracow, b4bronio@cyf-kr. edu.pl William Klink, Iowa, william-klink@uiowa . edu Kathrin Glantschnig, Graz, kathrin. glantschnig@uni-graz . at Leonid Glozman, Graz, leonid. glozman@uni-graz . at Victor Mandelzweig, Jerusalem, victor@helium. fiz .huj i.ac.il Alexander Molochkov, Dubna, alexm@thsun1. j inr. ru Hans-Christian Pauli, Heidelberg, paul i@mpi-hd. mpg. de Willibald Plessas, Graz, plessas@bkfug. kfunigraz .ac.at Milan Potokar, Ljubljana, milan.potokar@ij s.si Ica Stancu, Liege, fstancu@ulg .ac.be Michele Viviani, Pisa, michele ,viviani@pi . infn. it Robert Wagenbrunn, Graz, robert. wagenbrunn@pv .infn.it Bojan Golli, Ljubljana, boj an. goll i@ij s.si Damijan Janc, Ljubljana, damij an. j anc@ij s.si Rajmund Krivec, Ljubljana, raj mund. krivec@ij s.si Mitja Rosina, Ljubljana, mitj a . rosina@ij s.si Electronic edition http://www-f1.ijs.si/BledPub/ Contents Preface............................................................. V Effective interaction for the hyperspherical formalism Nir Barnea, Winfried Leidemann and Giuseppina Orlandini............... 1 Do we see the chiral symmetry restoration in baryon spectrum? L. Ya. Glozman ..................................................... 5 Coupled channel formulation of decays in the point form Goldstone boson exchange model W. Klink, S. Bofii, L. Glozman, W. Plessas, M. Radici, and R. Wagenbrunn ... 7 Quasilinearization method and its application to physical problems V B. Mandelzweig and R. Krivec ...................................... 10 On the structure of the pion: A QCD-inspired point of view H.C.Pauli.......................................................... 12 A three-body confinement force in constituent quark models Fl. Stancu and S.Pepin................................................ 24 Few-nucleon reactions of astrophysical interest M. Viviani, A. Kievsky, L.E. Marcucci, S. Rosati, and R. Schiavilla.......... 29 Covariant electromagnetic and axial nucleon form factors in the Goldstone-boson-exchange constituent quark model W. Plessas, R.F. Wagenbrunn, S. Boffi, L. Ya. Glozman, W. Klink, and M. Radici 32 Excitation of non-quark degrees of freedom in N* B. Golli, P. Alberto and M. Fiolhais..................................... 38 Constraints on effective constituent quark masses from phenomenology D. Janc and M. Rosina................................................ 42 Are there locally precise three-body wave functions? R. Krivec, V B. Mandelzweig and M. Ya. Amusia ........................ 45 Unified quark-quark and quark-antiquark interactions for the meson and baryon sector Mitja Rosina ........................................................ 50 Preface The series of mini-workshops at Bled, which started in 1987 with the workshop on Mesonic Degrees of Freedom in Hadrons, has established its own character of friendly but productive confrontation of ideas, such that we intend to organize similar workshops every year. It is now for the seventh time that a small group of enthusiasts met in this renowned holiday resort to clarify several open problems of common interest. The topics of this meeting ranged from few-quark problems, schematic models for hadronic and nuclear physics, effective interactions between constituent quarks, baryonbaryon and meson-meson effective interaction, semirelativistic few-body calculations, accurate few-body calculations, and quasilinearization method. The participants enjoyed a focused, intense discussion and critical confrontation of their results and ideas in a friendly atmosphere. Every participant had up to one hour time for his exposition which could be interrupted by questions and remarks, plus half an hour of general discussion. The advantage of such mini-workshops is the ease with which the participants sincerely acknowledge not only the successes, but also the weak points and open problems in their research. The rather diverse set of topics covered by a relatively small group of people did not, as one would perhaps expect, cause a breakup into smaller groups. On the contrary, the spirit of interdisciplinarity was abundant, criticism was relaxed and direct, and old friends retraced their paths through physics. The mini-workshop took place in Villa Plemelj, bequeathed to the Society of Mathematicians, Physicists and Astronomers by the renowned Slovenia mathematician Josip Plemelj. The beautiful environment of Lake Bled helped a lot to the cheerful atmosphere and optimism in the presentations; and the occasional inclement weather contributed to the patience for long afternoon discussions. Ljubljana, December 2001 M. Rosina R. Krivec B. Golli Workshops organized at Bled > What Comes beyond the Standard Model (June 29-July 9,1998) > Hadrons as Solitons (July 6-17,1999) > What Comes beyond the Standard Model (July 22-31,1999) > Few-Quark Problems (July 8-15, 2000) > What Comes beyond the Standard Model (July 17-31, 2000) > Statistical Mechanics of Complex Systems (August 27-September 2, 2000) > What Comes beyond the Standard Model (July 17-27, 2001) > Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25-28, 2001) Bled Workshops in Physics Vol. 2, No. 1 A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 1) Bled, Slovenia, July 7-14, 2001 Effective interaction for the hyperspherical formalism* Nir Barneaa**, Winfried Leidemannb,c and Giuseppina Orlandinib,c a The Racah Institute of Physics, The Hebrew University, 91904, Jerusalem, Israel b Dipartimento di Fisica, Universita di Trento, I-38050 Povo, Italy c Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Trento Abstract. The effective interaction method, traditionally used in the framework of an harmonic oscillator basis, is generalized to the hyperspherical formalism. It is shown that with the present method one obtains an enormous improvement of the convergence of the hyperspherical harmonics series in calculating ground state properties, excitation energies and transitions to continuum states. In general we would like to use the hyperspherical harmonics (HH) basis functions to solve the A-body Hamiltonian where m is the nucleon mass and Vy is the NN interaction. In practice, looking for the eigenvectors of H in terms of the HH expansion turns out to be a notoriously difficult task. Therefore, one usually has to introduce correlation functions in order to accelerate the convergence of the calculation [1-4]. In this work, however, we shall explore another possibility and instead of using correlation functions we shall use the method of effective interactions [5]. This approach is largely used in shell-model calculations [6,7], where the harmonic oscillator basis is used in a truncated model space. Instead of the bare NN interaction one uses effective interactions inside the model space. Defining P as the projection operator onto the model space and Q = 1 — P as the projection onto the complementary space, the model space Hamiltonian can be written as In general the effective interaction appearing in Eq. (2) is an A-body interaction. If it is determined without any approximation, then the model-space Hamiltonian provides a set of eigenvalues which coincide with a subset of the eigenvalues (1) eff P . (2) * Talk delivered by Nir Barnea. ** E-mail: nir@vms.huji.ac.il 2 Nir Barnea, Winfried Leidemann and Giuseppina Orlandini of the original full-space Hamiltonian, Eq. (1). However, calculation of the exact A-body effective interaction is as difficult as finding the full-space solution. It is therefore customary to approximate Veff by a sum of two-body effective interactions determined from a 2-body problem. In the effective interaction hyperspherical harmonic (EIHH) approach [9] the division of the total HH space in P and Q spaces is realized via the HH quantum number K (P(Q) space: K < (>)Kmax). The total Hamiltonian is written in hyperspherical coordinates, H = i(-A p(3) where p is the hyperradius and Ap contains derivatives with respect to p only. The grand-angular momentum operator K2 is a function of the variables of particles A and (A — 1) and of KA-2 the grand angular momentum operator of the (A — 2) residual system [8]. Then from the total Hamiltonian one can extract a "two-body" Hamiltonian of particles A and (A — 1) 1 K2 H2(P) = ^+VA(A_1} , (4) which, however, contains the hyperspherical part of the total kinetic energy. Since the HH functions of the (A — 2) system are eigenfunctions of KA-2 one has an explicit dependence of H2 on the quantum number KA-2 of the residual system, i.e. H2 —> H^-2 . Applying the hermitian version of the Lee-Suzuki method [5] to H2 one gets an effective Hamiltonian H2eff. The effective interaction Veff is obtained from 2m p2 This Veff replaces Vy in Eq. (3) when we project the solution on the P-space. This effective potential has the following property: Veff —> Vy for P —> 1. Due to the "effectiveness" of the operator the solution of the Schrodinger equation converges faster to the true one. The HH formulation leads to various advantages: (i) Veff itself is p dependent, therefore it contains some information on the "medium", (ii) because of the above mentioned KA-2 dependence the (A-2) residual system is not a pure spectator, and (iii) an additional confining potential is not needed, since the presence of p in Eq. (4) automatically confines the two-body system to the range 0 < rA_(A_!) < Vlp. We would like to point out that Veff(Kmax) can be viewed as a kind of momentum expansion, since the short range resolution is increased with growing Kmax. The convergence for the calculation of mean values can be improved introducing the corresponding effective operators. Of course for the calculation of the mean value of the Hamiltonian, i. e. Eb, one already makes use of an effective operator, namely H^Aff2. To this end let us present a simple example for the rate of convergence of the HH series with the effective interaction for ground state energy, and rms matter radii. In Fig. 1 we illustrate the convergence patterns with bare and effective interactions for binding energy and radius of 4He with the MTV potential. It is vrfr2(p) = H&-f2(P)-—— (5) lism 3 K Fig. 1. Binding energy (a) and root mean square radius (b) of the A=4 system for the Malfliet-Tjon potential MTV as a function of the hyperangular quantum number K. The asymptotic value has been indicated by a dashed line. readily seen that the effective interaction improves the convergence drastically. For the effective interaction one nearly obtains the correct values for energy and radius with a rather low K of 4. The convergence with the bare interaction is considerably worse since even with K = 20 one does not have completely converged results. Summing up, in this talk we have presented the recently developed hyper-spherical effective interaction method. In this approach the two-body effective interaction depends on the A-body hyperradius and on the state of the A-2 rest system explicitly. The EIHH method has been so far applied to few-body nuclei in the mass range A = 3 ^ 6, interacting through central and noncentral potential models including the realistic NN force AV14. Our results for these systems 4 Nir Barnea, Winfried Leidemann and Giuseppina Orlandini show that the method leads to an excellent convergence of the hyperspherical expansion. References 1. Yu. I. Fenin and V. D. Efros, Sov. J. Nucl. Phys. 15, 497 (1972). 2. see e.g. S. Rosati, A. Kievsky and M. Viviani, Proceedings of the XHIth European conference on Few-Body problems in physics, Sept. 9-14, Marciana Marina, Isola d'Elba, Italy, p. 563 (1992). 3. N. Barnea, W. Leidemann, and G. Orlandini, Nucl. Phys. A650, 427 (1999). 4. R. Krivec and V. B. Mandelzweig, Phys. Rev. A 42, 3779 (1990). 5. K. Suzuki and S. Y. Lee, Progr. Theor. Phys. 64, 2091 (1980). 6. M. Hjorth-Jensen, T. Engeland, A. Holt, and E. Osnes, Phys. Rep. 242, 37 (1994). 7. P. Navratil and B. R. Barrett, Phys. Rev. C 59,1906 (1999). 8. V. D. Efros, Yad. Fiz. 15, 226 (1972) [Sov. J. Nucl. Phys. 15,128 (1972)]. 9. N. Barnea, W. Leidemann and G. Orlandini, Phys. Rev. C 61, 054001 (2000). Bled Workshops in Physics Vol. 2, No. 1 A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 5) Bled, Slovenia, July 7-14, 2001 Do we see the chiral symmetry restoration in baryon spectrum? L. Ya. Glozman* Institute for Theoretical Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, One feature of QCD that is well understood is that the theory possess an approximate SU(2)L x SU(2)R symmetry and that this symmetry is spontaneously broken. As a consequence the properties of the low-lying hadrons are strongly affected by spontaneous breaking of chiral symmetry (SBCS). One striking feature of hadronic physics is the appearance of parity doublets for highly excited baryons (baryons with a mass of ~ 2 GeV and above). Recently it has been suggested that these parity doublets can be explained by an effective restoration of chiral symmetry for these highly excited states [1]. One of the experimentally established concepts, the quark-hadron duality, states that the experimentally observed spectral density p(s) at s —> oo is dual to the contribution of the free quark loop plus perturbative corrections (e.g. in e+ e- —> hadrons; in this case the "asymptotic" regime sets up at s ~ 2 — 3 GeV). However in perturbation theory there is no SBCS. Thus one sees immediately that SU(2)l x SU(2)R symmetry must be manifest in the spectrum at asymptotically high mass. Assuming that the process of chiral restoration is smooth (once one goes up in excitation energy from the ground state), as it indeed follows from the operator product expansion, one can understand the appearance of the systematical parity doublets in the upper part of baryon spectrum as a manifestation of chiral symmetry restoration [2]. One of the immediate implications is that the concept of constituent quarks, which may be adequate in the SBCS regime, becomes irrelevant high in the spectrum [1]. Effective chiral restoration implies that the physical baryon states high in the spectrum fall into multiplets of SU(2)L x SU(2)R. Constraint of parity conservation implies that such multiplets must be a direct sum of two irreducible representations: (1/2,0) ® (0,1/2), (3/2,0) ® (0,3/2) and (1/2,1) ® (1,1/2). The preceding representations are the only ones which contain no states with isospin greater than 3/2. Such I > 3/2 baryon states have never been observed. If the first two cases were realized in nature, then the spectra of highly excited nucleons and deltas would consist of parity doublets. However, the energy of the parity doublet in the nucleon spectrum a-priori would not coincide with the energy of the doublet with the same spin in the delta spectrum. This is because these doublets * E-mail: leonid.glozman@uni-graz.at Austria 6 L. Ya. Glozman would belong to different representations of SU(2)L x SU(2)R. On the other hand, if ( 1/2,1 ) © ( 1,1/2) were realized, then the highly lying states in N and A spectrum would consists of multiplets that contain one N parity doublet and one A parity doublet with the same spin and are degenerate in mass. We show that available spectroscopic data for nonstrange baryons in the ~ 2 GeV region is consistent with excited baryons approximately falling into (1/2,1) © (1,1/2) representation of SU(2)l x SU(2)r with approximate degeneracy between positive and negative parity N and A resonances of the same spin. References 1. L. Ya. Glozman, Phys. Lett. B475,329 (2000) 2. T.D. Cohen and L. Ya. Glozman, hep-ph/0102206, to appear in Phys. Rev. D Bled Workshops in Physics Vol. 2, No. 1 A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 7) Bled, Slovenia, July 7-14, 2001 Coupled channel formulation of decays in the point form Goldstone boson exchange model * W. Klinka**, S. Boffib, L. Glozmanc, W. Plessasc, M. Radicib, and R. Wagenbrunnc a University of Iowa, Iowa City, Iowa, USA b University of Pavia, Pavia, Italy c Institute for Theoretical Physics, Graz, Austria In the Goldstone Boson Exchange (GBE) model [1] the hyperfine interaction is generated by a constituent quark-meson vertex. Combined with a "semi relativis-tic" kinetic energy and linear confinement, eigenvectors of the three quark Hamil-tonian have been calculated [2]; the eigenvalues give a good fit for the low-lying nucleon and strange baryon spectrum. But the spectrum is a point spectrum, so that excited states have no widths; it is desirable to augment the GBE model to include strong and electromagnetic decays. The GBE Hamiltonian can be reinterpreted as a mass operator in point form relativistic quantum mechanics [3]. The goal of this note is to show how the mass operator on the three quark space can be enlarged to a mass operator on the direct sum of a three quark plus three quark and meson space, with transitions (off diagonal mass operator) between the two spaces produced by the original quarkmeson vertex. To show how the matrix mass operator is constructed it will prove useful to review some features of the point form. Recall that in the point form of relativistic quantum mechanics all interactions are in the four-momentum operator, which must satisfy the point form equations: Pv]= 0 (1) uaPhUX1 = (A-1);PV , (2) where Ua is a unitary operator representing the Lorentz transformation A. These equations are simply the Poincare commutations relations with global rather than infinitesimal Lorentz generators. The mass operator is given by M = %/P • P and must have a spectrum that is bounded from below. Multiparticle states such as the three quark states do not have nice transformation properties under Lorentz transformations; in particular the Wigner rotation of each particle is different, so that the spins and orbital angular momentum cannot be coupled together. Velocity states are multiparticle states in their overall * Talk delivered by W. Klink. ** E-mail: william-klink@uiowa.edu 8 W. Klink, S. Boffi, L. Glozman, W. Plessas, M. Radici, and R. Wagenbrunn center of momentum, boosted by a four-velocity v: |v,ki,m >: = UB(v)(|k ,H1 > -|kn>) (3) ^J|p! .CT! >...|pn,CTn >^0^ [Rwt), (4) where pi = B(v)kt,Y_ ki = 0, and RWi = B 1 (pt/m)B(v)B(ki/m). Under Lorentz transformations, using the definition, Eq.3, velocity states transform as UA|v, ki, p > = |Av,Rwki, p! > nDj ' (Rw); (5) Hi that is, the Wigner rotation RW = B-1 (Av)AB(v) is the same in all the arguments of the D functions, and is the same Wigner rotation multiplying all the internal momentum vectors, ki. That means all the spins as well as orbital angular momenta can be coupled together exactly as is done nonrelativistically [4]. From the relation between external and internal momenta, it also follows that Mfreelv,ki, pi > = Y_ ym? + ki2|v, ki,pi > (6) VH|v,ki,m > = vH|v,ki,m >, (7) where MfTee is the free mass operator, and VH is the free four-velocity operator. Then the free four-momentum operator is MfTeeVH and it is the free mass operator that is perturbed to introduce dynamics in the relativistic system. This so-called Bakamjian-Thomas procedure [5] is implemented in the point form by writing = MVH, where now M is the sum of free and interacting mass operators, M = MfTee + Mi. M takes the place of the Hamiltonian in nonrela-tivistic quantum mechanics and it is not hard to show that if M commutes with the four-velocity operator the point form Eq.1 is satisfied, while if M commutes with Lorentz transformations, the point form Eq.2 is satisfies. Consider now a mass operator on the direct sum space of the form M Mag V t V Mbq-M (8) where M3q = MfT + Mconf but does not include the hyperfine interaction. V is the mass operator generated by the meson-quark vertex [6], < v, ki, pi|V|v, ki, p! > = < v, ki, m|£i(0)|v, ki, pi > f(Am) (9) where Am is m — m and m (respectively m ) is the mass of the velocity state, as given in Eq.6. The interaction Lagrangian couples the quark to the meson and f (Am) is a form factor which is determined by the hyperfine potential. Then the GBE mass operator can be written as Mgbe = M3q + VMn-1 Vt where the last term gives the hyperfine mass operator. Mn is the meson propagator while M3Q_m = y/M2BE + k2 + ^m2 +k2. The goal now is to reduce the coupled channel problem to one involving only the 3Q space, such that MGBE has another term added to it which accounts Coupled channel formulation of decays ... 9 for the decays of the excited states: M|¥ > = m|¥ > (10) M3q|¥3q > +Vt|¥aq -M > = m|^3Q > V|^3Q > +M3q-mI^3q-m > = m|¥3Q-M > l^3q-m > =-r"|-V|¥3q > m - M3Q- M (M3Q+Vt--J-V)|¥3Q >=m|¥3Q > (11) m — M3Q-M (MGBE+Vt(--J--_L)V)|¥3Q >=m|¥3Q > (12) m — M3Q-M M„ The new term added on to MGbe represents the effect of the coupled channel; it contains the difference between the propagator from the four-particle space and the meson propagator. Since this last term is a difference between two operators, it is hopefully a small correction to the dominant term. If that is the case it should be possible to use perturbation theory to calculate the level shifts and widths of the excited states, which would then be small corrections to the levels given by the GBE mass operator itself. On the other hand if the correction is large, perturbation theory cannot be used. But then also the good fits to experimental data obtained by just the GBE mass operator will be significantly modified by adding on the new term so that it is no longer clear that any improvement results. References 1. L. Ya. Glozman, D. O. Riska, Phys. Rep. 268 (1996) 263. 2. L. Ya. Glozman, W. Plessas, K. Varga, R. F. Wagenbrunn, Phy. Rev. D58 (1998) 094030. 3. W. H. Klink, M. Rogers, Phys. Rev. C58 (1998) 3605. 4. W. H. Klink, Phys. Rev. C58 (1998) 3617. 5. B. Bakamjian, L. H. Thomas, Phys. Rev. 92 (1953) 1300; B. D. Keister, W. N. Polyzou, Adv. Nucl. Phys. 20 (1991) 225. 6. W. H. Klink "Constructing Mass Operators from Interaction Lagrangians", nucl-th/0012031. Bled Workshops in Physics Vol. 2, No. 1 A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 10) Bled, Slovenia, July 7-14, 2001 Quasilinearization method and its application to physical problems * V. B. Mandelzweiga** and R. Krivecb a Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel b J. Stefan Institute, P.O. Box 3000, 1001 Ljubljana, Slovenia The general properties of the quasilinearization method [1-3], particularly its fast convergence, monotonicity and numerical stability are analyzed and illustrated on different physical problems. The method approximates a solution of a nonlinear differential equation by treating nonlinear terms as a perturbation about linear ones, and is not based, unlike perturbation theories, on the existence of some kind of a small parameter. Each approximation of the method sums many orders of the perturbation theory. The method provides accurate and stable answers for any coupling strengths, including for super singular potentials for which each term of the perturbation theory diverges and the perturbation expansion does not exist even for a very small couplings. In order to further analyze and highlight the power and features of the quasilinearization method (QLM), we have made [2] numerical computations on the nonlinear ordinary first order differential equations for the S-wave scattering length a0 = a(oo) and phase shifts 60, respectively, obtained in the variable phase approach [4]. We have considered different singular and nonsingular, attractive and repulsive potentials, namely Yukawa, Poschl-Teller and Newton potentials, and have compared the results obtained by the quasilinearization method with the exact solutions. It is shown also [3] that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers. Our conclusions can be formulated as follows: i) The QLM treats the nonlinear terms as a perturbation about the linear ones [1] and is not based, unlike perturbation theories, on the existence of some kind of small parameter. As a result, as we see on our examples, it is able to handle, unlike the perturbation theory, large values of the coupling constant. ii) The method provides very accurate and numerically stable and fast convergent answers for any values of the coupling constant giving the accuracy of * Talk delivered by V. B. Mandelzweig. ** E-mail: victor@phys.huji.ac.il Quasilinearization method and its application 11 at least five significant figures required in this work. Already the first few iterations provide precise answers for small and intermediate values of the coupling constant. The number of iterations necessary to reach a given precision only moderately increases for larger values of the coupling constants. iii) The method provides very accurate and numerically stable answers also for any potential strength in the case of super singular potentials for which each term of the perturbation theory is infinite and the perturbation treatment is not possible even for a very small coupling. In view of all this, since most equations of physics, from classical mechanics to quantum field theory, are either not linear or could be transformed to a nonlinear form, the quasilinearization method may turn out to be extremely useful and in many cases more advantageous than the perturbation theory or its different modifications, like expansion in inverse powers of the coupling constant, the 1 /N expansion, etc. References 1. V. B. Mandelzweig, Quasilinearization method and its verification on exactly solvable models in quantum mechanics, J. Math. Phys. 40, 6266 (1999). 2. R. Krivec and V. B. Mandelzweig, Numerical investigation of quasilinearization method in quantum mechanics, Computer Physics Communications, 138, 69 (2001). 3. V. B. Mandelzweig and F. Tabakin, Quasilinearization Approach to Nonlinear Problems in Physics with Application to Nonlinear ODEs, Computer Physics Comm., 2001, in press. 4. F. Calogero, Variable Phase Approach to Potential Scattering, Academic Press, New York, 1965. Bled Workshops in Physics Vol. 2, No. 1 A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 12) Bled, Slovenia, July 7-14, 2001 On the structure of the pion: A QCD-inspired point of view H.C. Pauli* Max-Planck Institut fur Kernphysik, D-69029 Heidelberg, Germany Abstract. The effective interaction between a quark and an anti-quark as obtained previously with by the method of iterated resolvents is replaced by the so called up-down-model and applied to flavor off-diagonal mesons including the pion. The only free parameters are those of canonical quantum chromo-dynamics (QCD), particularly the coupling constant and the masses of the quarks. The so obtained light-cone wave function can be used to calculate the pion's form factor, particularly its mean-square radius can be computed analytically. The results allow for the exciting conclusion that the pion is built by highly relativistic constituents, in strong contrast to composite systems like atoms or nuclei with non-relativistic constituents. 1 Introduction One of the most urgent problems in contemporary physics is to compute the structure of hadrons in terms of their constituents, based on a covariant theory such as QCD. Among the hadrons the pion is the most mysterious particle. I have proposed an oversimplified model, the ^i.-model, which has many drawbacks but the virtue of being inspired by QCD and of having the same number of parameters one expects in a full theory: namely the 6 flavor quark masses, the strong coupling constant (7) and one additional scale parameter (8) originating in the murky depth of renormalization theory. The model is QCD-inspired by virtue of the fact, that it is based on the full light-cone Hamiltonian as obtained from the QCD-Lagrangian in the light-cone gauge, with zero-modes disregarded. In consequence, the pion is treated on the same footing as all other pseudo-scalar and pseudo-vector mesons. The model should be contrasted to Lattice Gauge Calculations, see for example [1]. It is not generally known that LGC's have considerable uncertainty to extrapolate their results down to such light mesons as a pion. It is also not generally known that lattice gauge calculations get always strict and linear confinement even for QED, where we know the ionization threshold. The 'breaking of the string', or in a more physical language, the ionization threshold is one of the hot topics at the lattice conferences [2]. Moreover, in order to get the size of the pion, thus the form factor, another generation of computers is required, as well as physicists to run them. * E-mail: pauli@mpi-hd.mpg.de On the structure of the pion: A QCD-inspired point of view 13 The model should be contrasted also to phenomenological approaches. They usually do not address to get the pion. For the heavy mesons, as well for light baryons [3], where they are so successful, phenomenological model have quite many parameters, in any case more that the above canonical ones. A detailed comparison and systematic discussion of the bulky literature can however be postponed, until we are ready to solve the full Eq.(11). The model should be contrasted, finally, to Nambu-Jona-Lasinio-like models which are so successful in accounting for isospin-aspects. I cannot quote the huge body of literature but mention in passing that the NJL-models are not renormal-izable, that NJL has no relation to QCD, and that NJL deals mostly with the very light mesons. There is no way to treat the heavy flavors, see also [4]. 2 Motivation The light-cone approach to the bound-state problem in gauge theory [5] aims at solving HlcW = M2|¥). If one disregards possible zero modes and works in the light-cone gauge, the (light-cone) Hamiltonian HLC is a well defined Fock-space operator and given in [5]. Its eigenvalues are the invariant mass-squares M2 of physical particles associated with the eigenstates |¥). In general, they are superpositions of all possible Fock states with its many-particle configurations. For a meson, for example, holds l^meson) = L ^qq , Ai)|qq) ¥gg (xi,ku, At)|gg) i i + L ¥qq g (xi, k_Li ,Ai)|qq g) + ¥qq qq (xi, k_Li ,Ai)|qq qq) + .... ii If all wave functions like ¥qq or ¥gg are available, one can analyze hadronic structure in terms of quarks and gluons [5]. For example, one can calculate the space-like form factor of a hadron quite straightforwardly. As illustrated in Fig. 1, it is just a sum of overlap integrals analogous to the corresponding non-relativistic formula [5]: F(q2) = Hefl [EI ft^ (*i>^(*i>:(1) Here ea is the charge of the struck quark, A2 > q 2, and i _ J kLi — xiqL + qL for the struck quark Li ~ \ kLi — xiqL for all other partons, with q 2 = Q2 = —q2. All of the various form factors of hadrons with spin can be obtained by computing the matrix element of the plus current between states of different initial and final hadron helicities. 3 The method of iterated resolvents Because of the inherent divergences in a gauge field theory, the QCD-Hamiltonian in 3+1 dimensions must be regulated from the outset. One of the few practical ways is vertex regularization [5,6], where every Hamiltonian matrix element, 14 H.C. Pauli particularly those of the vertex interaction (the Dirac interaction proper), is multiplied with a convergence-enforcing momentum-dependent function. It can be viewed as a form factor [5]. The precise form of this function is unimportant here, as long as it is a function of a cut-off scale (A). Perhaps one can attack the problem of diagonalizing the (light-cone) Hamil-tonian HLC by the discretized light cone quantization (DLCQ), see for example [7]. But, alternatively, it might be better to reduce the many-body problem behind a field theory to an effective one-body problem. The derivation of the effective interaction becomes then the key issue. By definition, an effective Hamiltonian acts only in the lowest sector of the theory (here: in the Fock space of one quark and one anti-quark). And, again by definition, it has the same eigenvalue spectrum as the full problem. I have derived such an effective interaction by the method of iterated resolvents [6], that is by systematically expressing the higher Fock-space wave functions as functionals of the lower ones. In doing so the Fock-space is not truncated and all Lagrangian symmetries are preserved. The projections of the eigenstates onto the higher Fock spaces can be retrieved systematically from the qq-projection, with explicit formulas given in [8]. x, k_L+ (1-x) q. p+q p+q Fig. 1. Calculation of the form factor of a bound state from the convolution of light-cone Fock amplitudes. The result is exact if one sums over all . p p Let me sketch the method briefly. Details may be found in [6,8]. DLCQ with its periodic boundary conditions has the advantage that the LC-Hamiltonian is a matrix with a finite number of Fock-space sectors, which we denumerate by n, with 1 < n < N. The so called harmonic resolution K = LP+/(2n) acts as a natural cut-off of the particle number. As shown in Figure 2, K = 3 allows for N = 8, and K = 4 for N = 13 Fock-space sectors, for example. The Hamiltonian matrix is sparse: Most of the matrix elements are zero, particularly if one includes only the vertex interaction V. For n sectors, the eigenvalue problem in terms of block matrices reads n 2ji|Hn(w)|j>(j|YM> = EM , fori = 1,2,... ,n. (2) j=i On the structure of the pion: A QCD-inspired point of view 15 n Sector 1 qq 2 gg 3 qqg 4 qqqq 5 ggg 6 qqgg 7 qqqq g 8 qqqqqq 9 gggg 10 qqgg g 11 qqqqgg 12 qqqq qqg 13 qqqqqqqq 1 qq • -< ■ ■ ■ 2 99 • • • • • • 3 qqg -c • • • 4 qqqq • • • -< • • 5 ggg • • • • 6 qqgg • • -< • • 7 qqqq g • • —^ • 8 qq qq qq • • • • • • • -< 9 9999 • • • • • 10 qqgg g • • • • • 11 qq qq gg • • • • • • > — 12 qq qq qq g • • • • • • • -c 13 qqqq qqqq • • • • • • • Fig. 2. The Hamiltonian matrix for a meson. The matrix elements are represented by energy diagrams. Only vertex diagrams V are shown. Zero matrices are marked by a dot I can always invert the quadratic block matrix of the Hamitonian in the last sector to define the n-space resolvent Gn, that is GnN = -¿-j^r. (3) W - HnM Using Gn, I can express the projection of the eigenfunction in the last sector by n-1 (n|¥(w)> = Gn(w) £ (n|Hn(w)|j) (j|¥(w)>, (4) j=i and substitute it in Eq.(2). I then get an effective Hamiltonian where the number is sectors is diminished by 1: Hn-1 (w) = Hn(w) + Hn(w)Gn(w)Hn(w). (5) This is a recursion relation, which can be repeated until one arrives at the qq-space. The fixed point equation E(w) = w determines then all eigenvalues. For the block matrix structure as in Figure 2, with its many zero matrices, the reduction is particularly easy and transparent. For K = 3 one has the following sequence of effective interactions: Hs = Ts, H7 = T7 + VGsV, H6 = T6 + VG7V, H5 = T5 + VG6V. (6) 16 H.C. Pauli The remaining ones get more complicated, i.e. H4 = T4 + VG 7 V + VG 7 VG 6 VG 7 V, (7) H3 = T3 + VG6V + VG6VG5VG6V + VG4V, (8) H2 = T2 + VG3V + VG5V, (9) Ht = Tt + VG3V + VG3VG2VG3V. (10) For K = 4, the effective interactions in Eq.(6) are different, see for example [8], but it is quite remarkable, that they are the same for the remainder, particularly Eq.(10). In fact, the effective interactions in sectors 1-4 are independent of K: The continuum limit K ^ 00 is then trivial, and will be taken in the sequel. In the continuum limit, the effective interaction in the qq-space has thus two contributions: A flavor-conserving piece Ueff-conser = VG3V and a flavor-changing piece Ueff-change = VG3VG2VG3V. The latter cannot get active in flavor-off-diagonal mesons. Notice that these expressions are an exact result. 4 The eigenvalue equation in the q q-space After some approximations [6], the effective one-body equation for flavor offdiagonal mesons (mesons with a different flavor for quark and anti-quark), becomes quite simple: M2(x, k± ; Ai ,A2|^) 1 m2 + kj2 m2 + kj2 + ■ X 1 — X ' dx'd2k| R(x',k[; A) ct (x, k±; Ai ,A2|^) (11) y/x{\ -x)x'(l -x') Q (Ai ,A2|S|A1 ,A2) (x', k|; Ai.A^). Here, M2 is the eigenvalue of the invariant-mass squared. The associated eigen-function ^ = ¥q£| is the probability amplitude (x, k^; Ai,A2|^} for finding a quark with momentum fraction x, transversal momentum k^ and helicity Ai, and correspondingly the anti-quark with 1 — x, — kj_ and A2. The mj and m2 are (effective) quark masses and ct is the (effective) coupling constant. The mean Feynman-momentum transfer of the quarks is denoted by Q2 = Q2(x,k±;x,,k[) =-- [(k! -ki)2 + (k2-k^)2] , (12) and the spinor factor S = S(x, k^; x', k_[) by (Ai,A2|S|A'1 ,\'2) = [u(ki,Ai)yiiu(k/1,A^)] [v(k^, A^vfe,A2)]. (13) The regulator function R(x', k[; A) restricts the range of integration as function of some mass scale A. I happen to choose here a soft cut-off (see below), in contrast to the previous sharp cut-off [9]. Note that Eq.(11) is a fully relativistic equation. I have derived the same effective interaction also with the method of Hamiltonian flow equations, see [10]. On the structure of the pion: A QCD-inspired point of view 17 The effective quark masses mi and m2 and the effective coupling constant a depend, in general, on A. In the spirit of renormalization theory they are renor-malization constants, subject to be determined by experiment, and hence-forward will be denoted by mi, m2, and a, respectively. In next-to-lowest order of approximation the coupling constant becomes a function of the momentum transfer, ct —> cc(Q; A), with the explicit expression given in [6]. 5 The Ti-model and its renormalization It might be to early for solving Eq.(11) numerically in full glory like in Ref.[9]. Rather should I try to dismantle the equation of all irrelevant details, and develop a simple model. The quarks are at relative rest, when k^ = 0 and x = x = mi /(mi + m2). For very small deviations from these equilibrium values the spinor matrix is proportional to the unit matrix, with (Ai,A2|S|AiA2> - 4mi m2 6a1,a; &A2,Ai , (14) for details see [10]. For very large deviations, particularly for kj2 ^ kj2, holds Q2 ~ kj2, and (TilS|Ti>- 2kj2. (15) Both extremes are combined in the "Ti-model" [10]: S 4m m2 4m m2 A2 qi = + + Wlth R(A>Q) = A^+Q2' (16) It interpolates between two extremes: For small momentum transfer, the '2' generated by the hyperfine interaction is unimportant and the dominant Coulomb aspects of the first term prevail. For large momentum transfers the Coulomb aspects are unimportant and the hyperfine interaction dominates. The model over-emphasizes many aspects: It neglects the momentum dependence of the Dirac spinors and thus the spin-orbit interaction; it also neglects the momentum dependence of the spin-spin interaction. But the 2 creates havoc: Its Fourier transform is a Dirac-delta function with all its consequences in a bound-state equation. Here is an interesting point: One is familiar with field theoretic divergences like the effective masses and the effective coupling constant. One is used less to "divergences" residing in a finite number 2. They must be regulated also, and renormalized. In consequence I replace Eq.(11) by M2^(x, k_L) a 3K2 m2 + kjj m2 + kj2 + ■ 1 x ^(x, kj) dx'd2k(_ (4mi m2 2A2 , , - -+ TT—7^7 PMX >kJ> (17) \/x{l -x)x'(1 -x') V Q2 A2 + Q2 x where^(x,kj) = (x,kjT,i |^>. 18 H.C. Pauli V \ \vvw \ \ \ \ \ \ \ \ \ \ \ ww \ Www X \ \ \ \\\ \ \ \ \ \ 1 1 1 1 1 1 1 1 1 1 1 1 1 \\\\ Nv \ \\\ \V\SS \\Vsw . \ VNXNSSS \ \ xvsnsns \ \ \ \Xs s' \ \ \ Wv Fig. 3. Nine contours an (A) are plotted versus A/A, from bottom to top with n = 4,3, • • • , -3, -4. The contours are obtained by M0 (a, A) = nA2 + M^. The thick contour n = 0 refers to the pion with M2 = M^. Mass unit is A = 350 MeV. Fig.4. The pion wave function ®(p) is plotted versus p/(1.338m) in an arbitrary normalization. The filled circles indicate the numerical results, the open circles the analytical function ®a (p). For equal quark masses mi = m2 = m, the eigenvalues depend now on three parameters, the canonical a and m, and the regularization scale A. The dependence can be quite strong as seen in Figure 3. There, the lowest mass-squared eigenvalue is plotted versus a and A for the fixed quark mass m = 406 MeV. Since A is an unphysical parameter, its impact must be removed by renor-malization. Recently, much progress was made on this question [11,12]: Adding On the structure of the pion: A QCD-inspired point of view 19 u d s c b u 768 871 2030 5418 d 140 871 2030 5418 s 494 494 2124 5510 c 1865 1865 1929 6580 b 5279 5279 5338 6114 Table 1. The calculated mass eigenvalues in MeV. Those for singlet-ls states are given in the lower, those for singlet-2s states in the upper triangle. to R(A, Q) acounterterm C(A, Q) and requiring that the sum R(A, Q) = R(A, Q) + C(A, Q), and thus M2(A;a,m), be independent of A, determines C(A, Q). One remains with R(A, Q) = |2/(|2 + Q2). In line with renormalization theory, one then can go to the limit A —> oo. In turn, |i becomes one of the parameters of the theory to be determined by experiment. 6 Determining the canonical parameters The theory has seven canonical parameters which have to be determined by experiment: a, | and the 5 flavor masses mf (if we disregard the top). How can we determine them? The problem is not completely trivial. Let me restrict first to the light flavors. With mu = md = m, one has 3 parameters, and in consequence needs 3 experimental data. The pion mass Mn = 140 MeV and the rho mass Mp = 768 MeV do not suffice. One needs a third datum, the mass of an exited pion, for example. Since the mass of the excited pion is not known with sufficient experimental precision, and since the Ti-model might be to crude a model to begin with, I choose here mu = md = 406 MeV and Mn* = Mp = 768 MeV, for no good reason other than convenience. These assumptions are less stringent than they sound, by two reasons. First, the rho has a mass less than 2m and should be a true bound state. Second, the Yukawa potential in Eq.(17) acts like a Dirac-delta function in pairing theory for example: it pulls down essentially one state, the pion, but leaves the other states unchanged. We thus remain with the two parameters a and Each of the two equations, M0(a, |) = Mn and M2(a, |) = M^* determine a function a(|). Their intersection point determines the required solution, which is a = 0.761 and | = 1.15 GeV [11]. These differ marginally from the previous analysis [10], with | = 1.33 GeV, for which Figure 3 yields a = 0.6904. Once I have the up and down mass, the strange, charm and bottom quark mass can be determined by reproducing the masses of the K,- D0 and B,- respectively. The parameters in the Ti-model can thus be taken as a | m- = md ms mc mb ^ 0.6904 1.33 GeV 406 MeV 508 MeV 1666 MeV 5054 MeV. ( ) 20 H.C. Pauli u d s c b u 768 892 2007 5325 d 140 896 2010 5325 s 494 498 2110 — c 1865 1869 1969 — b 5278 5279 5375 — Table 2. Empirical masses of the flavor-off-diagonal physical mesons in MeV. Vector mesons are given in the upper, scalar mesons in the lower triangle. Mff M Mexp 7T° 140 140 135 140 485 549 V 661 958 958 11c 2870 2915 2980 11b 8922 8935 — Table 3. Flavor-diagonal mass eigenvalues in the FM-model for pseudo-scalar mesons with the parameter a = (491 MeV)2. Mff M Mexp P° 768 768 768 CD 768 832 782 ® 973 1019 1019 J/V 3231 3242 3097 T 9822 9825 9460 Table 4. Flavor-diagonal mass eigenvalues in the FM-model for pseudo-vector mesons with the parameter a = (255 MeV)2. 7 The masses of the physical mesons Solving Eq.(17) with the parameters of Eq.(I) generates the mass2-eigenvalues of all flavor off-diagonal pseudo-scalar mesons. They are compiled in Table 1. The corresponding wave functions are also available, but not shown here. In view of the simplicity of the model, the agreement with the empirical values [13] in Table 2 is remarkable. The mass of the first excited states in Table 1 correlates astoundingly well with the experimental mass of the pseudo-vector mesons, as given in Table 2. Notice that all numbers in Tables 1 and 2 are rounded for convenience. Since the T^-model in Eq.(17) does not expose confinement one should emphasize that the difference between the physical meson masses in Table 1 and the sum of the bare quark masses is larger than a pion mass. One could call this a kind of practical confinement. On the structure of the pion: A QCD-inspired point of view 21 What about the flavor diagonal mesons?- They cannot be a solution to Eqs.(11) or (17), since the flavor-changing piece of the full effective interaction can generate matrix elements between different flavors. Thus far the precise structure of the flavor changing part Ueff-change = VG3VG2VG3V has not been analyzed in detail, because it requires a considerable effort. Rather, the following flavor-mixing model (FM-model [15]) has been investigated. In the FM-model, the full effective Hamiltonian including its flavor mixing is reduced to the lowest ff-states, i.e. to (2) i0 and the upper limit from imposing the right sequence in the spectrum, i.e. from the requirement Xs > Xi . The closer c is to the lower limit, the larger is the gap between the colour octet and singlet states. To see this, let us take c = -1.43. There is of course some arbitrariness in choosing m and fiw. As typical values for quark models we take m = 0.340 GeV and hw = 0.6 GeV [6]. This implies an octet-singlet gap AE = Es — E1 « 5.5 GeV. For c = 0 (no three-body force) one would have AE = « 2.1 GeV. Therefore the gap is increased considerably by a three-body force with a strong negative strength. This is a desired feature for quark models with three valence quarks (no gluons). In the same way one can show that the decuplet state is located above the octet with quite a large gap for a limiting value of c. Let us now consider tetraquarks, i.e. q2q"2 systems and denote the quarks by 1 and 2 and the antiquarks by 3 and 4. One can first form q q" pairs which are next coupled to colour singlets. These states are either singlet-singlet states |113124) or octet-octet states |813824) (see Ref. [5], chapter 10). One can have a three-body interaction acting in a q2q" subsystem as C ijk = - d abc Ff F? F: j 'k (13) c 26 Fl. Stancu and S.Pepin or a three-body interaction acting in a qq2 subsystem as Cijk = dabc F? F* F^ where Fi Af (14) (15) is the charge operator of an antiquark. The operators (13) or (14) between these tetraquark states have the same eigenvalue, which is calculated in two steps. First, one evaluates the eigenvalue between the states |312334} and 34), where the two quarks couple either to a 3 or a 6 state and the antiquarks to a 3 or a 6 state. One obtains -5/9 for the |312334} state and 5/18 for |612634}. The physically relevant states |113124 > and |813824 > are then defined by the transformations (see e.g. Ref. [7]): Thus one gets: and I13I24 >= Y ^|3i2334 > +Y -|6I2634 > , |8i3824 >= — ^■|3i2334 > +y^|6i2634 > . - 15 2 5 (I13I24IC123II13I24) oc [-(--) + - —] c = 0 - 2 5 15 5 ?13824|Cl23|8l3824) OC [-(--) + - — ] C = -— C , (16) (17) (18) (19) which shows that with a negative c one raises the expectation value of the octet-octet above the singlet-singlet state more than with c = 0. This implies that in the presence of a 3-body confinement interaction with c < 0 the coupling between octet-octet and singlet-singlet states due to a hyperfine splitting will be diminished, which amounts to make a ground state tetraquark less stable. This seems to be consistent with the experimental observation that no stable tetraquark system has been seen so far. The q6 systems are important for the NN problem. Here we discuss the sector IS = (01) or (10). It is well known that the physical NN state is a combination of three symmetry states containing the orbital configurations [6]O and [42] O, as shown for example in [8]. In fact the three symmetry states allowed by the Pauli principle can be combined into the NN and AA states and the unphysical hidden-colour CC state. The latter has an important role at short separation between the 3q clusters. Using 3-body fractional parentage coefficients, one can calculate the matrix elements of the three-body force (4)-(6) in the basis of the states |NN >, A three-body confinement force in constituent quark models 27 |AA > and |CC >. This gives rise to the following matrix: NN AA CC NN 28 8TC 38a/5 405 38a/5 135 C AA 38V5 405 C 121 405C 76 l35C CC 38a/5 135 C 76 l35C 9 5C The eigenvalues of this matrix are Ei = c/9,E2 = c/9 and E3 = 20c/9. This shows that the effect of the 3-body colour confinement on NN and AA is identical and rather small as compared to that on CC. In particular for a negative value of c, the spectrum of NN, AA and CC lowers and shrinks. For a positive c, the situation is the other way round. This means that, for c < 0, V3b itself brings some attraction and implies a stronger coupling of CC to NN and AA due to a hyperfine interaction. This will lead to a reduced hard core repulsion in the NN potential. In conclusion, a three-body confinement force can affect the spectrum of multiquark systems in a positive or negative way, depending on the strength c. In particular, if c is negative, the unphysical octet and decuplet states of a system of three quarks become well separated from the colour singlet states, which is a desired feature for models with three valence quarks only. In tetraquarks its role is also positive because it decouples the colour octet-octet state from the singlet-singlet one, the first being unphysical. For six-quark systems the role of the three-body confinement force with a negative strength is controversial. It increases the coupling of the hidden-colour CC states to the physical NN and AA states. On one hand, this brings more attraction into the NN potential, which is useful to lower too high hard core potentials, but on the other hand this implies stronger Van der Waals forces. The latter is in contrast with the hopes of Ref. [4]. Details of the calculations can be found in Ref. [9]. One should notice that the present study is based on a simple harmonic oscillator confinement. It would be useful to extend it to a more realistic confinement. Also, for tetraquarks and six-quark systems the results are derived for compact configurations, i.e. for zero separation between the hadronic clusters, here of type q q" or q3. It would certainly be interesting to study non-zero separation distances (molecular type configurations). Acknowledgements: We are most grateful for the useful comments made by Mitja Rosina regarding the role of the three-body forces in tetraquarks and six-quark systems. References 1. J. Carlson, J. Kogut and V. R. Pandharipande, Phys. Rev. D27 (1983) 233. 28 Fl. Stancu and S.Pepin 2. A. E. Dorokhov and N. I. Kochelev, preprint JINR-E2-86-847, 1986; M. Oka and S. Takeuchi, Nucl. Phys. A524 (1991) 649; S. Takeuchi and M. Oka, Phys. Rev. Lett. 66 (1991) 1271. 3. T. T. Takahashi, H. Matsufuru, Y. Nemoto and H. Suganuma, Phys. Rev. Lett. 86 (2001) 18; C. Alexandrou, Ph. de Forcrand and A. Tsapalis, hep-lat/0107006. 4. V. Dmitrasinovic, Phys. Lett B499 (2001) 135. 5. Fl. Stancu, Group Theory in Subnuclear Physics, Oxford University Press, 1996. 6. Fl. Stancu, S. Pepin and L. Ya. Glozman, Phys. Rev. C56 (1997) 2779; C59 (1999) 1219 (erratum). 7. D. M. Brink and Fl. Stancu, Phys. Rev. D49, 4665 (1994) 8. M. Harvey, Nucl. Phys. A352 (1981) 301; ibid. A481 (1988) 834 (erratum). 9. S. Pepin and Fl. Stancu, Phys. Rev. D, to be published. Bled Workshops in Physics Vol. 2, No. 1 vTLv A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 29) Bled, Slovenia, July 7-14, 2001 Few-nucleon reactions of astrophysical interest * M. Viviania**, A. Kievskya, L.E.Marcuccib, S. Rosatib, and R. Schiavillac a INFN, Sezione di Pisa, Pisa, Italy b Physics Department, University of Pisa, Pisa, Italy c Jefferson Lab and Old Dominion University, Virginia, USA Abstract. The study of few-nucleon reactions at low energies has evolved into a mature area of research. We present a brief review of the methods and recent advances obtained by the Pisa-Jefferson Lab /ODU collaboration. One of the most important applications of nuclear physics is to make precise model-independent predictions of rates of astrophysical reactions, with possibly quantitative estimates of "theoretical" uncertainties. This goal is particularly important for providing nuclear physics input to stellar models. In fact, for several of these reactions the rates are too small to be measured in laboratories and therefore their estimates can be obtained only by using a theoretical model. The description of these processes requires the knowledge of the initial (bound) and final (in general, continuum) nuclear states and the use of electromagnetic (EM) and weak current operators constructed consistently with the interaction used to generate the wave functions. The nuclear EM and weak current models developed so far (see, for example, Refs. [2,3]) include one- and two-body operators. The EM two-body current is constructed consistently with the nucleon-nucleon interaction, in order to satisfy current conservation [4]. The vector part of the weak current is obtained from the isovector EM current via the CVC hypothesis. The axial part has been constrained by reproducing the experimental value of tritium beta decay. The present model of the current has been tested in numerous few-nucleon processes and it is thought to be quite realistic. The A = 3,4 bound and continuum wave functions have been calculated by expanding on a basis of correlated hyperspherical harmonic (CHH) functions [1]. Such a technique has been proven to be rather accurate. It should be emphasized that with the CHH technique the inclusion of the Coulomb potential, clearly very important in the energy range considered here, does not present any difficulty. Let us review briefly the CHH method for the trinucleon bound state. The wave function has been written as (1) * Talk delivered by M. Viviani. * E-mail: michele.viviani@pi.infn.it 30 M. Viviani, A. Kievsky, L.E. Marcucci, S. Rosati, and R. Schiavilla -rk, yi =(rj+rk-2ri)/V3) (2) where P = V x2 + y2 ' cos^t = xi/p, (3) rt denoting the position of particle i, and the variable p is the so-called hyperradius. The correlation functions f «(rjk) have been introduced in order to accelerate the convergence, as explained below. The angle-spin-isospin functions Y«(jk,i) are defined as Ya(jk,i) = {[YU (Si) ® Yl„ (yt)]Aa j ® sals«J j ® • (4) Each a-channel is specified by the angular momenta 1«, La coupled to give Aa, and by the spin (isospin) sj« (tj) and s« (t«) of the pair jk and the third particle i, coupled to give Sa (T). The antisymmetrization of the wave function ¥ requires 1« + s«k + tj be odd; in addition, £a + La must be even for positive parity states and odd for negative ones. The hyperspherical polynomials (2)pK",L" are given by [5] (2)Pk toi) = la(sin^i)L«(cos^i)1«pn«+1/2>1«+1/2(cos2^t) , (5) where N^0"^ is a normalization factor and P£'b a Jacobi polynomial. The grand orbital quantum number is given by K = £a + La + 2n, n being a non-negative integer. In Eq. (1), K0 = 1« + La is the minimum grand orbital quantum number and Ka is the maximum selected value. In the limit Nc , Ka —» oo, the expansion basis used in Eq. (1) is complete. In a practical calculation, Nc and Ka are kept finite and increased until convergence in the binding energy is achieved. In this regard, it should be stressed the importance of the inclusion of the correlation functions fa(rjk) in Eq. (1) for improving the convergence rate. The standard (uncorrelated) HH expansion is recovered if such functions are set equal to one. However, for potentials containing a strong repulsion at small distances, the rate of convergence of the HH expansion is found to be very slow. The role of the correlation function f«(r) in Eq. (1) is therefore to accelerate the convergence of the expansion by improving the description of the system when two particles are close to each other. These correlation functions are obtained from solutions of suitable two-body zero-energy Schrodinger equations, with a technique outlined in ref. [1]. The present approach has been applied also to the the study of the ground state of the a particle and to the continuum states of A = 3,4 nuclear systems. Some of the computed S-factors are presented in Table 1. The effect of the two-body currents is seen to increase significantly as the number of particles involved in the reaction grows. The last column of the table shows the "theoretical uncertainty", namely the variance of the S-factor values obtained by choosing different realistic nuclear Hamiltonians H for calculating the bound and scattering wave functions. This choice is reflected also in the current. Indeed, the EM current j(q)-more precisely, its longitudinal part-is constructed to satisfy the continuity x rj Few-nucleon reactions of astrophysical interest 31 reaction Si(0) Si+2(0) AS/S V + d -> 3 He + y p + 3He —> 4He + e+ +-ye 3.90 x 10~2b 3.95 x 10~2b 0.5% 1.32 x 10~7 1.88 x 10~7 2% 29.0 x 10~23 9.64 x 10~23 5% Table 1. Zero energy S-factors (in Mev-barn) for various reactions. In the column labelled S i (0) (S i + 2(0)), the S -factors obtained including the one-body (one- and two-body) terms in the nuclear current are reported. In the last column, the "theoretical uncertainty" (obtained as explained in the text) is also reported. equation q • j (q) = [H, p(q)] and therefore depends on the nuclear interaction chosen to describe the nuclear states. The axial weak current too is related to H via PCAC and also because one of its parameter is fixed by reproducing the tritium beta decay rate within a given model Hamiltonian. These theoretical uncertainties are rather small, indicating that the dependence on the interaction model is rather weak. Recently, another model for the transition current, obtained from an effective field theory (EFT) based on the chiral symmetry, has been developed [6]. Using this approach, the transition operators are organized according to a "sys-tematic"power counting in the heavy baryon chiral perturbation theory. They have been worked out up to N3LO: at this level the transition operators have one and two body terms and there appears one unknown parameter in the chiral Lagrangian, which has been fixed again by fitting the tritium p-decay rate [7]. With this model for the transition current, and using the nuclear wave functions calculated as above, the S-factors of the first and third reaction considered in Table 1 turn out to be rather close to those calculated here. For example, Si+2(0) for the reaction p + p 2H+e+ + ve is 3.94 x 10-25 [7], only 0.25% smaller than the result reported in Table 1. This substantiates the confidence on the accuracy of the theoretical estimates of these S-factors. The extension to other processes, as muon capture on light nuclei, is in progress. References 1. A. Kievsky, S. Rosati, M. Viviani, Nucl. Phys. A577, 511 (1994); Few Body Systems 18, 25 (1995) 2. J. Carlson and R. Schiavilla, Rev. Mod. Phys. 70, 743 (1998) 3. L. E. Marcucci et al., Phys. Rev. C63, 015801 (2000) 4. D. O. Riska, Phys. Scr. 31, 471 (1985) 5. M. Fabre de la Ripelle, Ann. Phys. (NY) 147, 281 (1983) 6. T.-S Park et al, Nucl. Phys. A596, 515 (1996) 7. T.-S Park et al, nucl-th/0106025 Bled Workshops in Physics Vol. 2, No. 1 vTLv A Proceedins of the Mini-Workshop Selected Few-Body ... (p. 32) Bled, Slovenia, July 7-14, 2001 Covariant electromagnetic and axial nucleon form factors in the Goldstone-boson-exchange constituent quark model * W. Plessas**a, R.F. Wagenbrunn*** a, S. Boffib, L.Ya. Glozmana, W. Klinkc, and M. Radicib a Institut fiir Theoretische Physik, Universitat Graz, A-8010 Graz, Austria b Dipartimento di Fisica Nucleare e Teorica, Universita di Pavia, I-27100 Pavia, Italy and INFN, Sezione di Pavia, I-27100 Pavia, Italy c Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA Abstract. We show that within the Goldstone-boson-exchange constituent quark model not only the light and strange baryon spectra but also electromagnetic and axial nucleon form factors come out in rather good agreement with phenomenology. A consistent description of the electroweak nucleon form factors is essentially dependent on the inclusion of Lorentz boost effects which can be treated accurately in the point form approach to relativistic quantum mechanics. 1 GBE constituent quark model A promising approach to low-energy hadrons consists in constituent-quark models (CQMs). Starting from rudimentary attempts more than two decades ago, one has constantly improved the description and gained a lot of insight into the properties of hadrons at low and intermediate energies. Evidently, CQMs can at most be effective models of quantum chromodynamics (QCD) in a domain where the fundamental theory is not (yet) accurately solvable. It appears that below a certain energy scale in the light-flavor sector the spontaneous breaking of chiral symmetry (SBxS) of QCD is responsible for the generation of constituent quarks as quasiparticles with dynamical masses much greater than the corresponding current-quark masses. Numerous theoretical as well as experimental evidences hint to a chirally broken phase of QCD. If one understands the generation of constituent quarks as being caused by SBxS, one should at the same time also accept the other consequences of SBxS. According to that one is left with a residual SU(3)V symmetry associated with the existence of Goldstone bosons. It follows that one is dealing with new degrees of freedom in the light-flavor sector of QCD, namely with constituent quarks and Goldstone bosons instead of current quarks and gluons, which represent the * Talk delivered by W. Plessas and R.F. Wagenbrunn. ** E-mail: plessas@bkfug.kfunigraz.ac.at *** E-mail: Robert.Wagenbrunn@uni-graz.at Covariant electromagnetic and axial nucleon form factors ... 33 original degrees of freedom in the QCD Lagrangian. Following this idea, already several years ago, Glozman and Riska have suggested a quark-quark hyperfine interaction in light and strange baryons which is mediated by Goldstone boson exchange (GBE) [1]. This interaction has a spin-flavor symmetry, which is obviously most appropriate for describing the spectra of light and strange baryons. Evidently, in addition to this hyperfine interaction one has to advocate some confining forces. Here we specifically adhere to the version of the GBE CQM published in Ref. [2]. It comes with a mutual quark-quark interaction Vij = Vconf + VX) (1) with a confinement potential in linear form Vconf (rij )= Vo + Crij (2) and the chiral interaction consisting of only the spin-spin part of the pseudoscalar meson exchange Vx(rtj) "3 7 2 Y_ V7t(rij)A[A[+^ VKlnJXltf +V11(rij)A?Af + -Vv(rtj) .F=1 F=4 (3) Here ot are the Pauli spin matrices and At the Gell-Mann flavor matrices of the individual quarks. For all details of the parameterization of the Vn)K,n,n' and the values of all parameters of the model we refer to the original paper Ref. [2]. The complete three-Q Hamiltonian of the model is then given by H = H0el + V (4) where H0el is the relativistic kinetic-energy term 3 irel io i=1 HToel = L A2 + m?> (5) with mt the masses and kt the 3-momenta of the constituent quarks, which are restricted by Y.3=1 k = 0. The phenomenological spectra of the light and strange baryons are reasonably well described by the model. In Fig. 1 we show here just the spectra of N and A. The complete spectra including all the strange baryons can be found in Ref. [2]. CT i • CT I' 2 Electromagnetic and axial nucleon form factors For the calculation of electromagnetic and axial nucleon form factors one has to deal with boosts from the rest frame in which the nucleon wave function is originally obtained to a moving frame since at least one, the incoming or outgoing nucleons is moving in the scattering process. In any CQM the intrinsic movement of the constituent quarks inside a nucleon is highly relativistic making a 34 W. Plessas, R.F. Wagenbrunn, S. Boffi, L.Ya. Glozman, W. Klink, and M. Radici M [MeV] 1800 -1700 -1600 -1500 1400 1300 1200 11001000 -900 m N ~~i-r 1+ i- 2 2 —I-1— 3+ 3~ 2 2 5 + 2 3 + 2 Fig. 1. N and A spectra of the GBE CQM. The bars are the calculated masses of states with angular momentum and parity JP. The shaded boxes are the experimental masses. nonrelativistic treatment of boosts obsolete. Even for very low momenta transferred to the nucleon from the external probe (electron or neutrino) constituent quarks with large momenta are involved. One can incorporate relativity into a quantum theory with a finite number of degrees of freedom (as a CQM) by utilizing relativistic Hamiltonian dynamics (i.e. Poincare-invariant quantum mechanics) [3]. From the various (unitarily equivalent) forms that are possible when defining the (kinematic) stability subgroup [4] the point form has some obvious advantages [5] in our studies. In this form the four-momentum operators PH contain all the dynamics. They commute with each other and can be diagonalized simultaneously. All other generators of the Poincare group are not affected by interactions. In particular, the Lorentz generators are interaction-free and make the theory manifestly covariant. Through the introduction of so-called velocity states [6] we can carry out all necessary transformations of the momentum dependences in the wave functions and the relativistic quark spin rotations associated with boosting the nucleon state in an accurate manner. In order to come from the semirelativistic Hamiltonian as given in section 1 to relativistic Hamiltonian dynamics one can apply the so-called Bakamjian-Thomas (BT) construction [7] where all dynamical generators of the Poincare group are obtained from one auxiliary operator. This mass operator consists of a free part, which is just the relativistic kinetic energy, as given in Eq. (5), and an interaction term. The potential of the GBE CQM satisfies all conditions required for the relevant operators to fulfill the Poincare algebra. Even if the potential term Covariant electromagnetic and axial nucleon form factors ... 35 is nonrelativistic one can thus arrive at a relativistic quantum theory by reinterpreting the Hamiltonian as a mass operator in a BT construction. For the calculation of electromagnetic form factors it can be shown that the electromagnetic current operator can be written in such a way that it transforms as an irreducible tensor operator under the strongly interacting Poincare group. Thus the electromagnetic form factors can be calculated as reduced matrix elements of such an irreducible tensor operator in the Breit frame. The same procedure can be applied to the axial current. The current operator is assumed to be a single-particle current operator for point-like constituent quarks. This corresponds to a relativistic impulse approximation but specifically in point form. It is called the point-form spectator approximation (PFSA) because the impulse delivered to the nucleon is different from that delivered to the struck constituent quark. The electromagnetic current matrix elements have the usual form for a point-like Dirac particle, i.e. (pi.Aij^pi.Ai) = etu (pi.Ai^ulpi.AO, (6) with u(pt, At) the Dirac spinor of quark i with charge et, momentum pt, and spin projection At. Such a is not conserved a-priori but one can always construct a conserved current = — qH(q • j/q2), with q the 4-momentum transfer. The new added term does not affect the = 0,1,2 components which are used to calculate the form factors. The axial current matrix elements have the form (pi, A(|Aaipi, Ai) = u (pi, Ai) + 2fn q 2+m 9«qqk y5^Tau(pi,Ai), (7) where mn is the pion mass, fn = 93.2 MeV the pion decay constant, and Q2 = — q2, with q = pt — pt the momentum transferred to a single quark. The quark axial charge is assumed to be gA = 1, as for free bare fermions, and gnq is identified with the pion-quark coupling constant, with a numerical value as employed in the GBE CQM of Ref. [2]. Along this formalism we have calculated the complete set of electromagnetic and axial form factors, i.e. the electric and magnetic proton and neutron form factors as well as the axial and induced pseudoscalar nucleon form factors. The results have been published in a series of papers [9]. For example we show in Fig. 2 the ratios of the electric proton form factor to the dipole parameterization and to the magnetic proton form factor. The latter has recently been determined experimentally in a direct measurement at Jefferson Lab [8]. The data start to fall below one for momentum transfer squared Q2 > 1 GeV2. Obviously this tendency is reproduced by the PFSA result though the agreement with the data is not perfect. However, in a completely nonrelativistic impulse approximation (NRIA) calculation the ratio would be 1 for all Q2. In general the shapes of all electromagnetic form factors calculated in PFSA are in good agreement with the data up to Q2 - 1 GeV2. We mention, however, that for the magnetic form factors there are some small discrepancies from the data even at very low momentum transfers; of course, this affects also the magnetic moments. 36 W. Plessas, R.F. Wagenbrunn, S. Boffi, L.Ya. Glozman, W. Klink, and M. Radici A AA O Andivahis V Walker □ Sill A Hoehler O Bartel Gp / G„ = ci|R> + c2at|N) , (8) where àt is the creation operator for the lowest vibrational mode. The coefficients ct and the energy are determined by solving the generalized eigenvalue problem in the 2 x 2 subspace. The solution with the lowest energy corresponds to the Roper, while the orthogonal combination to one of the higher excited states with nucleon quantum numbers, e.g., the N (1710), provided the c-meson mass is sufficiently small. The energy of the Roper is reduced (see Table 1), though the effect is small due to weak coupling between the state (3) and the lowest vibrational state with the energy £1 . The reduction becomes more important if the mass of the c-meson is decreased. The energy of the combination orthogonal to the ground state is close to En + £1 with c-meson vibrational mode as the dominant component. mt En 2s-1s AEr AEr* c2 £1 1200 1269 446 354 353 0.05 1090 700 1249 477 367 364 0.12 590 Table 1. For two ff-masses we show the nucleon energy (EN), the Roper-nucleon energy splittings calculated from the single particle energy difference (2s-1s), the state (3) (AEr) and the state (8) (AER„). All energies are given in MeV. The electromagnetic nucleon-Roper transition amplitudes as well as the transition amplitudes to higher excitations with nucleon quantum numbers represent an important test which may eventually distinguish between the models listed at the beginning. The transverse helicity amplitude is defined as A 1/2 -z 2na kw d3r (R+i M llem(r) • e+i eik r|N 4,Mt (9) where kW is the photon momentum at the photon point, and the scalar helicity amplitude as 1/2 z 2na kw dr (R + 4-,MTl'em Jem (r) eikr|N + 7, M (10) Excitation of non-quark degrees of freedom in N* 41 Here J^m is the EM current derived from the Lagrangian density (1): jSn(l+qt+(7t x 9H7t)° • (11) The amplitudes (9) and (10) contain a phase factor, Z, determined by the sign of the decay amplitude into the nucleon and the pion. The new term in (8) does not contribute to the nucleon-Roper transition amplitudes. Namely, for an arbitrary EM transition operator O involving only quarks and pions we can write (N|a iO |N) = (N|[a i,O]|N) = 0, (12) because of (5) and since the operators an commute with O. A possible way to identify such a state would be to search for those excited states for which the amplitudes are strongly reduced compared to those calculated in a model with only quark degrees of freedom. There have been several attempts to calculate these amplitudes in various models, such as in chiral quark models [9,10], models with explicit gluon degrees of freedom [5] and relativistic versions of the constituent quark model [11]. Unfortunately, the present status of theoretical prediction is rather unclear because of a strong sensitivity of transition amplitudes on small variation of the Roper wave function. To understand the nature of the Roper remains a big challenge for theoreticians as well as for experimentalists. References 1. L. Ya. Glozman, Z. Papp, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. C 57 3406 (1998); W. Plessas, R. F. Wagenbrunn et al., these proceedings 2. P. A. M. Guichon, Phys. Lett. B 163, 221 (1985); Phys. Lett. B 164, 361 (1985) 3. W. Broniowski, T. D. Cohen and M. K. Banerjee, Phys. Lett. B 187, 229 (1987) 4. O. Krehl, C. Hanhart, S. Krewald, and J. Speth, Phys. Rev. C 62, 025207 (2000) 5. Zhenping Li, V. Burkert, Zhujun Li, Phys. Rev. D 46, 70 (1992); E. Carlson and N. C. Mukhopadhyay, Phys. Rev. Lett. 67 3745 (1991) 6. M. C. Birse, Prog. Part. Nucl. Phys. 25,1 (1990); T. Neuber, M. Fiolhais, K. Goeke and J. N. Urbano, Nucl. Phys. A 560, 909 (1993) 7. M. Fiolhais, B. Golli and S. Sirca, Phys. Lett. B 373, 229 (1996); L. Amoreira, P. Alberto, M. Fiolhais, Phys. Rev. C 62, 045202 (2000); A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 588, 801 (1995); A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 609, 488 (1996) 8. B. Golli and M. Rosina, Phys. Lett. B 165, 347 (1985); M. C. Birse, Phys. Rev. D 33,1934 (1986) 9. P. Alberto, M. Fiolhais, B. Golli, and J. Marques, Phys. Lett. B 523, 273 (2001), hep-ph/0103171 10. K. Bermuth, D. Drechsel and L. Tiator, Phys. Rev. D 37, 89 (1988) 11. S. Capstick, Phys. Rev. D 46, 2864 (1992); S. Capstick and B.D. Keister, Phys. Rev. D 51, 3598 (1995); H. J. Weber, Phys. Rev. C 41, 2783 (1990); F. Cardarelli, E. Pace, G. Salme and S. Simula, Phys. Lett. B 397, 13 (1997); F. Cano and P. Gonzalez, Phys. Lett. B 431, 270 (1998) Bled Workshops in Physics Vol. 2, No. 1 vTLv Proceedins of the Mini-Workshop Selected Few-Body ... (p. 42) Bled, Slovenia, July 7-14, 2001 Constraints on effective constituent quark masses from phenomenology * D. Janca** and M. Rosinaa>b a J. Stefan Institute, PO Box 3000,1001 Ljubljana, Slovenia b Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box 2964,1001 Ljubljana, Slovenia Abstract. From the assumption of a two-particle Hilbert space for mesons and from rather general properties of the effective quark-quark potential we constrain considerably the choice of effective constituent quark masses. 1 Nonrelativistic models We consider the following form of the Hamiltonian for the quark-antiquark system p2 H = — + V0(r) + cti • ct2Vs(tu1 ,m2;r), 2| where | is the reduced mass of the system and mi and m2 are the quark and antiquark masses. We make rather general assumptions about the potential: 1. The central potential V0(r) is flavour independent 2. The central potential is monotonic function or r and satisfies the conditions for a positive Laplacian and concavity d 2dV0 d2V0 —rz—— > 0 and ——7- < 0. dr dr dr2 3. The spin-spin potential Vs satisfies the condition that |Vs decreases with total mass of both quarks M = m1 + m2 4. The spin-spin potential for vector mesons is a monotonic function of r and has positive Laplacian —r2— 0 dr dr In the family of potentials which satisfy the conditions 1. and 2. one can find the "QCD inspired" Coulomb-plus-linear potential and power law potential a V0(r) =--+ |3r + U0, r V0(r) = A + Brp, Talk delivered by D. Janc. E-mail: damjan.janc@ijs.si Constraints on effective constituent quark masses ... 43 while the conditions 3. and 4. are satisfied for example by Vs(mi ,m2; r) a e -t/to mi m2 r a, r0 > 0. From these assumptions one can obtain inequalities between quark masses and masses of ground states of pseudoscalar and vector mesons, which to some extent restrict masses of constituent quarks as shown in Fig(1). c 1950P b = 5250MeV b = 52 0 0MeV b = 515 0MeV 1900 1850 1800 1750 1700h ..................... 300 400 500 600 700 I u > 2 8 0MeV I u > 34 0MeV Fig. 1. Allowed mass region for strange and charmed quarks for different choices of light and bottom quarks. s 2 Semirelativistic models For heavy quark Q - light (or heavy) antiquark q pseudoscalar mesons we use the semirelativistic model with the Hamiltonian H = J p2 + tUq + J p2 + m2 + ) Vss + V, v v mq where we assume that the expectation value of F(mQ) Vss/mq is a monotonically decreasing function of mq and that both Vss and V are flavour (mass) independent. The Hamiltonian for all vector mesons in our model has the general form H = y P2 + mf + y P2 + m2 + V (mi,m2), where we demanded that the expectation value of V(mi ,m2) is a decreasing function of the quarks masses from where it follows that E(K*)— E(p)