Understanding the origin of the fermion families and of the structure of fermion masses and mixings is one of the outstanding open questions in Particle Physics. In the Standard Model (SM), all the fermion masses are generated through Yukawa couplings, after the spontaneous breakdown of the SU_L (2) U_y (1) SU (3)_c gauge symmetry into . However, in the SM the family structure of Yukawa couplings is not constrained by gauge symmetry and, as a result, fermion masses and mixings are arbitrary. There have been various attempts at understanding the pattern of quark masses and mixings, including the systematic search for texture zeros [1], where fermion mass matrices, in an appropriate weak-basis, are assumed to have some zero entries at the unification scale. A somewhat different approach, proposed by some of us [2], is the hypothesis of Universal Strength of Yukawa (USY) couplings. The idea is motivated by the observation that among the couplings of the SM, only Yukawa couplings may be complex, all other couplings are constrained to be real by hermiticity of the Lagrangian. This led to the suggestion that all Yukawa couplings have universal strength, with all the family structure being contained in their phases. By studying the chiral limit, where the first generation of quarks is massless, we have been able to construct specific ansätze [3] based on USY, where all the four parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix are correctly predicted in terms of quark mass ratios. The extension of these ideas to the leptonic sector is actively pursued (see section on Neutrino Physics).

[1] P. Ramond, R. G. Roberts and G.G. Ross, Nucl. Phys. B406 (1993), 19.

[2] G. C. Branco, J. I. Silva-Marcos and M. N. Rebelo, Phys. Lett. B237 (1990) 446.

Neutrino physics has received an increased attention throughout the world after the discovery of the oscillations of atmospheric neutrinos announced last year by the Super-Kamiokande collaboration. Hints for neutrino oscillations have been accumulating in the solar and atmospheric neutrino data during the last two decades, but it is only now that a firm evidence for these oscillations has been obtained. Neutrinos can only oscillate if they are massive, so the discovery of neutrino oscillations implies the discovery of a neutrino mass. Since in the standard model of electroweak interactions neutrinos are massless, evidence for neutrino oscillations is the first firm evidence of new physics, i.e. physics beyond the standard model.

Neutrinos play a very special rôle in particle physics. The smallness of their masses is commonly believed to be due to the existence of new physics at very large energy scales, not accessible to present-day and probably even to future accelerators. Therefore, by studying neutrino masses one can obtain a very important information about new physics.

Our group has been involved in the analysis of possible patterns of neutrino mass matrices and particle physics mechanisms and models capable of producing the requisite masses and mixings of neutrinos for a number of years (see, e.g., [1-6]. In the next three years we are planning to continue and extend these studies. The topics of the future work include three main, distinct but related, directions:

I.2.1 Analysis of the experimental data

The main purpose of the analysis of the experimental data obtained by the solar and atmospheric neutrino experiments as well as of neutrino data from reactor and accelerator experiments is to acquire information on neutrino parameters (such as masses, mixing angles and magnetic moments) from the available and forthcoming experimental data. Recently, very important data on both solar and atmospheric neutrinos were obtained by the Super-Kamiokande experiment [7-9]; the experiment is still running, and more and more accurate data will become available in the future. In addition, a few new solar neutrino experiments (SNO, Borexino) and reactor and accelerator long-baseline neutrino experiments (Kamland, K2K, CERN-Gran Sasso) are expected to start operating in the next few years. It is therefore very important to analyze the available data and make predictions for the forthcoming experiments for various currently allowed regions of the neutrino parameter space. Analyses of this kind should include calculating the probabilities of oscillations of solar and atmospheric neutrinos as well as reactor and accelerator neutrinos. An important ingredient of these studies is taking into account the effects of the medium on neutrino oscillations inside the Sun and inside the Earth (Mikheyev-Smirnov-Wolfenstein effect [10,11]). Another interesting mechanism of neutrino conversion that may be responsible for the experimentally observed deficit of solar neutrinos is neutrino spin precession in the solar magnetic field [12]. Analysis of this effect which is currently under way may give an important information on both the neutrino magnetic moment and the internal structure of the magnetic field of the Sun.

The main goal of extracting the information from various neutrino experiments is to fully reconstruct the neutrino mass matrix, i.e. to acquire a coherent pattern of all neutrino masses and mixing angles. The present-day data has already constrained the allowed structure of the neutrino mass matrix in a significant way, but a large degree of arbitrariness still remains. It is known that the neutrino mass differences possess a certain hierarchy, but it is not known if the masses themselves have such a hierarchy. Also, some of the neutrino mixing angles remain still uncertain. One particularly attractive possibility is that neutrinos are almost degenerate, i.e. their masses are nearly the same, although the mass differences may have some hierarchy. In this case neutrinos can constitute the so-called "hot dark matter of the universe" -- an important ingredient of the cosmological dark matter. The models with such neutrino spectrum are now being extensively studied in our group. One interesting feature of these models that neutrinos may have quite non-trivial CP-violating properties even if they are exactly degenerate in mass. CP violation can manifest itself in neutrino oscillations as a difference between the probabilities of oscillations of neutrinos and antineutrinos. Such effects are usually small when neutrino mass squared differences have a large hierarchy. Recent solar neutrino data, however, indicate that this hierarchy may be not too large, so the CP violating effects can be observable in the long-baseline neutrino oscillations experiments. Therefore the studies of the CP properties of neutrinos in various models which are currently being performed in the group are, in our opinion, of considerable interest.

One of the most attractive mechanisms of neutrino mass generation which gives very natural explanation of the smallness of the neutrino masses is the so-called seesaw mechanism. The implications of this mechanism for the structure of neutrino mass matrix is currently being studied in the group. This includes the possibility of having degenerate or quasi-degenerate neutrino mass spectrum as well as a more usual hierarchical mass spectrum. In particular, the possibility of obtaining the phenomenologically viable neutrino mass textures (special structures) is being investigated.

Our next direction of work includes studying neutrino mass models that are capable of producing the neutrino mass matrices consistent with the experimental observations. After the structure of the neutrino mass matrix is established, one can look for underlying particle physics models that can reproduce this mass matrix. This is actually the ultimate goal of the neutrino physics studies. Even though the exact form of the neutrino mass matrix is not known yet, the existing data constrain it in a very significant way. This allows one to discriminate between possible particle physics models of neutrino mass. One approach to the neutrino mass models is based on the so-called universal-strength Yukawa couplings model which was developed in our group and proved to be very fruitful. We are planning to continue the studies along these lines. Another approach which we are planning to pursue is to look for certain symmetries of Yukawa interactions which can explain the structure of neutrino mass matrices. Usually, these symmetries are imposed at some very high energy scale (typically, Grand Unification scale); obtaining the low-energy predictions of such models includes therefore running the corresponding Yukawa couplings down in energies. This is done with the use of the renormalization group equations (RGE). We are planning a RGE study of some particle physics models ofneutrino mass.

Collaborators: Research scientists: E. K. Akhmedov, G. C. Branco; J. Pulido, M. N. Rebelo; J. I. Silva-Marcos. Graduate students: F. Joaquim; A. M. Teixeira.

[1] G.C. Branco, L. Lavoura, M.N. Rebelo, Phys. Lett. 180B (1986) 264.

[2] G.C. Branco, W. Grimus, L. Lavoura, Nucl. Phys. B312 (1989) 492.

[3] G.C. Branco, M.N. Rebelo, J.I. Silva-Marcos, Phys. Lett. B428 (1998) 136.

[4] G.C. Branco, M.N. Rebelo, J.I. Silva-Marcos, Phys. Rev. Lett. 82 (1999) 683.

[5] J.I. Silva-Marcos, Phys. Rev. D59 (1999) 091301.

[6] J. Pulido, Zhi-Jian Tao, Phys. Rev. D51 (1995) 2428.

[7] The Super-Kamiokande Collaboration (Y. Fukuda et al.), Phys. Rev. Lett. 81 (1998) 1562.

[8] The Super-Kamiokande Collaboration (Y. Fukuda et al.), Phys. Rev. Lett. 82 (1999) 1810.

[9] The Super-Kamiokande Collaboration (Y. Fukuda et al.), e-Print Archive: hep-ex/9812011.

[10] S.P. Mikheyev, A.Yu. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913.

[11] L. Wolfenstein, Phys. Rev. D17 (1978) 2369.

[12] E.Kh. Akhmedov, Phys. Lett. B213 (1988) 64.

Among the possible extensions of the Standard Model a special attention in our group has been given to supersymmetry. In particular, we have been studying supersymmetric models that break R-parity. This can be achieved in two ways. In the first, R-parity is a good symmetry at the Lagrangian level but it is spontaneously broken [1]. The Nambu Goldstone boson of this breaking, the so called Majoron J, plays an important rôle in the strategy for the search of the experimental consequences of the models. In fact, this massless particle interacts very weakly with matter and escapes detection. The model has a very rich phenomenology because we need to have lepton number violation for its consistency. Therefore, we have predictions not only for the accelerators [2], but also for lepton number violating processes [3]. Also, the neutrinos acquire masses and their mixing angles can be calculated [1,4]. The other way of breaking R-parity is explicitly at the Lagrangian level. Although one can argue that this a less constrained scenario it is certainly an open possibility. If the explicit breaking occurs via a bilinear term in the superpotential of the form e _i L_i H_u , then possible problems with proton decay can be avoided and we have shown that this case is, for most purposes, an effective case of the previous spontaneous breaking model [5]. For the next three years we will continue to study these models. We will focus our attention in the following topics:

We will continue the study of Supersymmetric Unification with radiative breaking of R-parity. In particular, we want to study in a more rigorous way the neutral scalars spectrum. In this work the self-energies of the scalars will be correctly included. Also, the issues of the so called a _s problem will be addressed. As it is well known, there is a problem in the MSSM in reproducing the present data on a . It is also known that the introduction of threshold effects on the running of the Renormalization Group Equations (RGE) makes things better for the model. We want to make a systematic study to see if the bilinear R-parity model makes a further contribution in solving this problem of the MSSM. Preliminary calculations seem to indicate that this is indeed the case.

After the Super-Kamiokande data on the atmospheric neutrinos a renewed interest appeared in the study of mechanisms for the generation of neutrino masses. The models with spontaneously broken R-parity have already been investigated in this respect, but they do not constrain very much the parameters. In the explicit case with bilinear violation we have a situation were we are very close to the Minimal Supersymmetric Standard Model (MSSM) with just a few extra parameters (three if we consider the full three generation case). Then we can further constrain the model by imposing N=1 Supergravity unification, like in the constrained MSSM. Then we are in a unique situation to calculate the one loop radiatively generated masses for the neutrinos. Such a calculation is now under way.

Within the models with breaking of R-Parity (spontaneous and explicit) we will continue the study of signals for new physics at the LEP2 and LHC. For this we have upgraded our Monte Carlo event generator to those energies. Preliminary studies indicate that the most promising processes are those with two leptons plus missing energy in the final state, specially if one of the leptons is the tau lepton. At these energies the effects of R-parity in the production (single production of supersymmetric particles) will be very much suppressed. Then the difference with respect to the MSSM will be in the decay modes.

Like in all models where there is lepton number violation, the contributions to the very well measured process m ® e g and to the anomalous magnetic moments must be calculated. They will provide a very stringent test of the theory. We are interested in study the amount of non universality that is allowed at the GUT scale in order to obey the experimental limits for those processes. We are considering the full three generation problem with particular ansatzes for the quark and lepton mass matrices. The full set of RGE's is considered. This is the thesis project of Daniel F. Carvalho and the study of the MSSM is almost finished. Next we will consider the bilinear R-parity model.

Another topic focuses on FCNC reactions involving B⁰ decays which offer an opportunity for discovering indirect effects of new physics. At present there are only a few FCNC processes which have been observed experimentally, but the situation will change considerably in the foreseeable future due to the B factories presently under construction. In this project, we investigate exclusive and inclusive b® d e⁺e^- in MSSM employing the formalism developed before [6], with special emphasis on CP-violating effects .

Collaborators: J. C. Romão, F. Krueger, P. Nogueira, D. F. Carvalho, A. M. Teixeira.

[1] J. C. Romão and J. W. F. Valle, Nucl. Phys B381 (1992) 87; J. C. Romão, C. A. Santos and J. W. F. Valle, Phys. Lett. B288 (1992) 311.

[2] P. Nogueira, J.C. Romão and J. W. F. Valle, Phys. Lett 251B (1990) 142; J. C. Romão, F. de Campos, and J. W. F. Valle, Phys. Lett. B292(1992) 329; M. C. Gonzalez-Garcia, J. C. Romão and J. W. F. Valle,Nucl. Phys, B391 (1993) 100; A. Lopez-Fernandez, J. C. Romão, F. de Campos and J.W.F. Valle, Phys. Lett. B312 (1993) 240; J. C. Romão, F. de Campos, M. A. Garcia-Jareño, M. B. Magro and J. W. F. Valle, Nucl. Phys. B482 (1996) 3.

[3] J.C. Romão, N. Rius and J.W.F. Valle, Nucl. Phys. B363 (1991)369.

[4] J. C. Romão and J. W. F. Valle, Phys. Lett B272 (1991)436.

[5] M. A. Diàz, J C. Romão, José W.F. Valle, Nucl. Phys. B524 (1998) 23.

[6] F. Krueger and L.M. Sehgal, Phys.Rev. D55 (1997) 2799, F. Krueger and L.M. Sehgal, Phys.Rev. D56 (1997) 5452.

Violation of the discrete symmetry CP in the mixing of and has been discovered in 1964 [1]. There was no decisive experimental development on the subject The situation is now, however, in the midst of a major experimental revolution. On the one hand, the KTeV Collaboration [2] has annouced a measurement of which brings the Fermilab result on this parameter in agreement with an earlier measurement by the NA31 Collaboration at CERN [3]. This constitutes the first experimental observation of direct CP violation-i.e., of CP violation in decays and eliminates from consideration theoretical models of the superweak type [4].On the other hand, the first measurements of CP violation outside of the system will be performed at the B factories by the end of 1999.

The latter experiments will allow an unprecedented precision in the determination of the parameters of the Cabibbo-Kobayashi-Maskawa mixing matrix [5]. The measurement of the angle b will be almost immediate; on the contrary, the extraction of a will be obscured by the effects of penguin diagrams ("penguin pollution"). These effects may be estimated using SU(3) symmetry, as was first pointed out by Silva and Wolfenstein [6]. The consistency among the various measurements will allow one to probe the possible existence of physics beyond the Standard Model [7].

Recently, Meca and Silva [8] have pointed out that the transitions may be used to search for new-physics effects in mixing. We shall continue our investigation of new observables possibly generated by the existence of the parameters x introduced by these authors [9].

An outstanding theoretical issue is the so-called "strong CP problem", for which some of us have already invented a variety of possible solutions [10, 11]. We feel that more and simpler solutions may be found in the context of real CP violation [12]- i.e., of spontaneous CP breaking through real vacuum expectation values-as has been first hinted at by Lavoura [11]. We shall pursue the search for such solutions.

Another issue is theoretical models of CP violation based solely on neutral-scalar exchange. Although it is generally believed that such models are invariably of the superweak type, and are thus eliminated from consideration by the recent measurements of this may not be so. We have a wide experience on this kind of models [13], and shall try to demonstrate that they are able to evade the existing experimental bounds.

We have worked extensively on these subjects. Following our writing of a book [14], which has allowed us to identify some of the open questions in this field, we shall continue our investigations, trying to illuminate the dark points in the theoretical understanding of the fascinating phenomenon of CP violation.

Collaborators: Gustavo Castelo Branco, Frank Krüger, Luís Lavoura, João Paulo Silva, M.N. Rebelo and P.A. Parada

[1] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Physical Review Letters, 13, (1964) 138.

[3] NA31 Collaboration (G. D. Barr et al), Physics Letters B317, (1993), 233.

[4] L.Wolfenstein, Physical Review Letters 1381964) 562.

[5] N. Cabibbo, Physical Review Letters 10(1963) 652; M. Kobayashi and T. Maskawa, Progress of Theoretical Physics 49 (1973) 652.

[6] J. P. Silva and L. Wolfenstein, Physical Review D49 (1994) 1151. See also M. Gronau, J. L. Rosner, and D. London, Physical Review Letters 73 (1994) 21; N. Deshpande and X.-G. He, Physical Letters B336 (1994) 471 and Physical Review Letters 74 (1995) 26.

[7] Y. Nir and D. Silverman, Nuclear Physics 345B (1990) 301; Y. Nir and H. R. Quinn, Physical Review D42(1990) 1473; G. C. Branco, T. Morozumi, P. A. Parada, and M. N. Rebelo, Physical Review D48(1993) 1167 and Physics Letters B306 (1993) 398; J. P. Silva and L. Wolfenstein, Physical Review D55 (1997) 5331.

[8] C. C. Meca and J. P. Silva, Physical Review Letters 81 (1998) 1377.

[9] A. Amorim, M. G. Santos, and J. P. Silva, Physical Review D59 056001 (1999).

[10] L. Bento, G. C. Branco, and P. A. Parada, Physics Letters B267 (1991) 95; L. Lavoura, Physics Letters B391 (1997) 441.

[11] L. Lavoura, Physics Letters B400 (1997) 152.

[12] A. Masiero and T. Yanagida, hep-ph/9812225.

[13] L. Lavoura, International Journal of Modern Physics A9 (1994) 1873; G. C. Branco, W. Grimus, and L. Lavoura, Physics Letters B380 (1996) 119.

[14] G. C. Branco, L. Lavoura, and J. P. Silva, (to be published in July 1999 by Oxford University Press).

For high values of the Higgs boson mass , the interactions among the Higgs and gauge bosons become strong. Typical strong-interaction phenomena, as for example bound states and bootstraps, have already been studied in a simplified calculation limited to the Higgs and its self interactions only, using the homogeneous Bethe-Salpeter (BS) equation [1]. However, for a realistic description of the scalar sector of the Standard Model, it is necessary also include the longitudinal components of the gauge bosons. Therefore, an inhomogeneous, coupled-channel BS equation will have to be solved for the HH W⁺W^-, and ZZ channels. The search for resonances in this system requires an analytic continuation of the equations into the second Riemann sheet, employing contour-rotation techniques.

Collaborators: Prof.Eef van Beveren of the University of Coimbra.

[1] G.Rupp, Phys. Lett. 288, (1992) 99.

The work we propose to do, in the continuation of what we have previously done, will be centered on the following areas:

The Haldane-Shastry model was previously studied using the technique of exact diagonalization of small systems [1]. The properties at finite temperatures were studied diagonalizing completely the Hamiltonian and the properties of the ground state and of the first excited state were studied using the modified Lanczos method, taking into consideration the symmetries of the problem [2]. The energy spectrum shows, for S=1, the existence of the Haldane gap, as expected, extending to a system with long range interactions the results previously obtained by other authors for systems with short range interactions. We considered also a non frustrating interaction, introducing a model with alternating (antiferromagnetic nearest neighbors and ferromagnetic next nearest neighbors) and of long range interactions, decaying as a power law. We found a gapless spectrum for an integer spin S=1, which shows that these models define a new class different from the class considered by Haldane [3]. We intend to study now spin chains with oscillating interactions of the RKKY type, resulting from the coupling of the chain spins with the metallic matrix. In particular, we intend to study the influence of the conduction electron band-filling in the type of spectrum of the spin chain. One should expect a transition between the regimes with and without gap, due to the variation of the electron density.

Collaborators: Nathalie Guihéry, Laboratoire de Physique Quantique, Université Paul Sabatier, Toulouse, France.

We have studied the influence of the anisotropy in the interactions between antiferromagnetic chains on low energy excitations of these systems. Contrarily to what happens in the isotropic situation, we established that the systems with Ising interactions always have a gap for any (even or odd) number of chains [4]. We intend to study now the influence of anisotropy in coupled chains of the zig-zag type and to study recently proposed incommensurability effects. We also intend to study this model at finite magnetization in order to study charge dynamics in coupled chains of the t-J model, relevant for the study of the normal state of some high T_C superconductors.

Collaborators: Nathalie Guihéry, Laboratoire de Physique Quantique, Université Paul Sabatier, Toulouse, France; J.P. Rodriguez, Instituto de Ciencia de Materiales, C.S.I.C, Madrid, Spain and Department of Physics and Astronomy, California State University, Los Angeles, U.S.A.

The nature of the quasi-particle excitations of strong type II superconducting systems under the influence of strong magnetic fields was studied using the mean field theory. The density of states shows that a regime with gapless excitations (or with a very small gap) prevails at low temperatures and for strong magnetic fields. Criteria for the construction of a perturbative method, having as zero order approximation the high field limit, previously solved analytically [5], were established. The influence of random pinning of the vortices in the excitations of this type of superconductors is such that even in the presence of disorder there are gapless modes in the same high magnetic field regime, suggesting a topological nature for these excitations [6]. It was also proposed that the Hall conductivity could be used as order parameter to detect the transition between the regimes with and without gap in the superconductors of type II, when lowering the magnetic field [7]. We intend to study now in detail the transition between the two regimes.

Collaborators: Z. Tesanovic, Department of Physics and Astronomy, The Johns Hopkins University, U.S.A.

Recently, several electronic systems presenting non conventional behavior have been found. The n-channel Kondo lattice model at the level of one impurity shows critical behavior at low temperatures. The extension to the most interesting case of a lattice of n-channel Kondo impurities was studied, searching for anomalous behavior at low temperature. Using the 1/n expansion at the level of the non-crossing approximation (NCA) it is found that the behavior is similar to that of a single impurity and that the non-Fermi liquid behavior prevails. The system behaves as an incoherent metal (due to the coupling between the conduction electrons and the impurity). At low temperatures the lattice coherence should prevail and the system must show an instability. Summing the ladder diagrams we have found the temperature at which the instabilities show up in the spin and charge susceptibilities [8], giving rise to ordered states at low temperatures, as experimentally observed. We intend to study now in detail the different phases at low temperatures and, in particular, study the coexistence of antiferromagnetism and superconductivity recently observed experimentally.

Collaborators: M.A.N. Araújo, Departamento de Física, Universidade de Évora, Portugal; N.M.R. Peres, Departamento de Física, Universidade de Évora, Portugal.

Using the exact solution given by the Bethe-ansatz and the representation of the eigenstates of the Hubbard model in terms of the pseudoparticles, we have studied transport properties of one dimensional conducting systems, such as the conductivity as a function of frequency for T=0º and the Drude peak at finite temperature, clarifying several points less correct or not understood in the existing literature [9,10,11]. We intend to study now the several spectral functions of the Hubbard model and establish experimental signatures for the elementary excitations previously described in terms of pseudoparticles.

Collaborators: D. Baeriswyl, Institut de Physique Théorique, Université de Fribourg, Switzerland; D.K. Campbell, Department of Physics, University of Illinois at Urbana-Champaign, U.S.A; J.M.P. Carmelo, Departamento de Física, Universidade de Évora, Portugal; J. Lopes dos Santos, Departamento de Física, Universidade do Porto, Porto, Portugal; N.M.R. Peres, Departamento de Física, Universidade de Évora, Portugal.

We have previously done the study of the spin J coherent states [12,13], the application of these states to the construction of path integrals for quantum spins [14], and the analytical [14,15] and numerical [16] application of these techniques to simple problems. We intend now to apply these techniques to the study of non trivial problems involving systems of coupled spins or in contact with heat baths.

[1]- P.D. Sacramento and V.R. Vieira, Z. Phys. B 101, (1996) 441.

[2]- P.D. Sacramento and V.R. Vieira, J. Phys. Cond. Matt. 70, (1995) 8619.

[3]- P. D. Sacramento and V.R. Vieira, J. Phys. Cond. Matt. 9, (1997) 10687.

[4]- J. P. Rodriguez, P.D. Sacramento and V.R. Vieira, Phys. Rev.B 56, (1997) 13685.

[5]- Z. Tesanovic and P.D. Sacramento, Phys. Rev. Lett. 80, (1998) 1521.

[6]- P. D. Sacramento, Phys. Rev. B (1/4/1999).

[7]- P.D. Sacramento, preprint.

[8]- P.D. Sacramento and V.R. Vieira, Phys. Rev. B 58, (1998) 11119.

[9]- J.M.P. Carmelo, N.M.R. Peres and P.D. Sacramento, Condensed Matter Theories, vol. 13, (Nova, 1998) 277.

[10]- N.M.R. Peres, P.D. Sacramento, D.K. Campbell and J.M.P. Carmelo, Phys. Rev. B (15/3/1999).

[11]- J.M.P. Carmelo, N.M.R. Peres and P.D. Sacramento, preprint.

[12]- V.R. Vieira and P.D. Sacramento, J. Phys. A 27, (1994) L783.

[13]- V.R. Vieira and P.D. Sacramento, Ann. Phys. 242, (1995) 188.

[14]- V.R. Vieira and P.D. Sacramento, Nucl. Phys. B 448, (1995) 331.

[15]- V.R. Vieira and P.D. Sacramento, Proceedings of International Conference on Mathematical Physics and Stochastic Analysis, World Scientific.

[16]- V.R. Vieira and P.D. Sacramento, Physica A 207 (1994) 584.

III .1.1 Chiral symmetry, the non-perturbative regime of QCD, and hadrons

The main goal is to obtain a microscopic description, progressively closer to QCD, of the hadronic (and nuclear) forces based on quarks, both at low and high temperatures. Two main problems have to be addressed simultaneously: the non-perturbative nature of QCD chiral symmetry and quark loops. The strategy of the attack must therefore unfold along several directions, which must be made consistent both with the requirements of chiral symmetry (Ward identities) and the underlying gauge invariance (Wilson loops in the non-perturbative regime of QCD). With the progressively improved quark-quark effective interactions one is obtaining, we can then readdress hadronic problems like electromagnetic form factors, strangeness contents of hadrons, and N-N forces [1]. This in turn will shed phenomenological light on the approximations made, like for instance the cummulant expansion and the neglect of quark loops, that is, strong decay and hadronic loops. The presence of coupled channels [2] adds another barrier to an already extraordinarilycomplicated situation, heralding the need to go beyond BCS when dealing with the mechanism of spontaneous chiral-symmetry breaking. The effect of coupled channels is very important in the scalar sector and constitutes in its own right a research field (see the section entitled Relativistic Unitarised Quark Model). It is clear that, even at low energies, chiral symmetry and the non-abelian nature of QCD imply the existence both of a quark sea and multigluon configurations. These degrees of freedom are in some sense encoded in the chiral and gluon condensates. But there is a situation known under the name of heavy-ion physics (see secton Heavy ion Physics), where these degrees of freedom become "explicit", and these phenomena have attracted a lot of attention, both experimental and theoretical. The main reason for this renewal of interest in this interface zone between high-energy physics and nuclear physics stems from the possiblity-predicted a long tim e ago, but only now testable-of the existence of a new state of matter resulting from the fusion of hadronic matter. This quark-gluon plasma can occur precisely when two heavy ions (e.g. Pb-Pb) collide at SPS (CERN) energies. Portugal has had a remarkable rôle in this experimental area, participating in the NA38 and NA50 experiments, which for the first time seem to suggest the existence of this new state of matter. It should be noted that the verification of the quark-gluon plasma is a direct testing of QCD. Thus it is quite important as a test of our understanding of strong interactions. Our task in this area has mainly concentrated on building new verification methods for quark-gluon-plasma formation, testing these in particular against the results of the NA50 experiment. The objective has been so far, and will remain so, to be able to distinguish the formation of a dense hadron gas from a quark-gluon plasma.

It is clear that, when we deal with light quarks, chiral-symmetry breaking and confinement are the two relevant facts. On the one side, Nambu-Jona-Lasinio models, as well as instantons, explain the main physics of chiral-symmetry breaking, but do not have the colour-confinement property of QCD. On the other side, the confining models that try to explain light-quark phenomenology are mainly of two types: 1) models with a scalar confining interaction that explicitly breaks chiral symmetry, but ensures the spin-orbit interaction known in the heavy-quark limit; 2) models with a vector confining interaction which are chirally symmetric and allow spontaneous breaking of chiral symmetry, but are then in trouble with the spin-orbit interaction. The heavy-light interaction which we propose is cleanly obtained from QCD in a gauge-invariant approach and under the assumption of dominance of the bilocal correlator. It should be able to reproduce the expected confining linear potential and the non-perturbative spin-orbit coupling in the heavy-quark limit, and should also be chirally symmetric in the chiral limit [7]. For heavy-light systems, chiral symmetry and confinement appear to be strongly related, as also suggested by lattice simulations at non-zero temperature, where confinement and chiral-symmetry breaking show up at the same temperature [8]. It provides a physical picture very different from the traditional Dyson-Schwinger methods where chiral-symmetry breaking is associated with the (gauge-dependent) non-linear dynamics of a single quark [9]. But it also differs from the traditional, phenomenologically motivated, Dirac equations used in the literature for describing heavy-light systems in the no-recoil limit. In those works, a scalar confining interaction which explicitly breaks chiral symmetry is introduced by hand and phenomenologically justified. Here, a chirally non-invariant binding interaction naturally emerges on the chirally broken physical vacuum. The main pieces of the puzzle seem to go to the right place. A privileged situation where to test these ideas is provided by the heavy-light quark sector of QCD. The study of heavy-light-quark bound states has absorbed in the last years a huge amount of effort, both on the experimental side, where a lot of new data have been produced (compare, for instance, with [10] ), and on the theoretical side, due to the use of heavy-quark symmetries [11]. Some approaches resort to a calculation via phenomenological potential models [13,14], sum rules [15], or phenomenological relativistic equations. A unified description of the heavy-heavy and the heavy-light dynamics in the no-recoil limit was suggested on the basis of theso-called Stochastic Vacuum model of QCD [17]. This consists in a Gaussian approximation of the QCD gluodynamics, which is assumed to be governed by the two-point (non-local) gluon condensate. This assumption is confirmed by lattice-simulation data in the case of quasi-static quarks. In this way a new dynamical scale explicitly appears: the gluon correlation length T_g. Under this condition two different situations occur. If is bigger than all the other scales of the problem, then the one-body limit of the heavy-quark potential is recovered with the famous Eichten-Feinberg-Gromes spin-orbit, scalar-like interaction. The other case, with smaller than all the other scales of the problem, is the typical heavy-quark sum-rule situation. Realistic heavy-light systems are characterised by . In this case the quark propagator cannot be considered as a free one, and its dynamics is inherently nonperturbative. For m = 0, the QCD Lagrangian is chirally symmetric. Then, the most appropriate approach is to determine the physical chirally broken vacuum of the problem. This is done by means of the Bogoliubov-Valatin variational method.

In quark models, hadrons are traditionally described as bound states of valence quarks held together by some confining potential. Even in modern, QCD-inspired approaches, which treat current quarks as relativistic Dirac particles subjectto chiral-symmetry breaking, effects from strong decay and hadronic loops are usually neglected. One of the few exceptions is the chiral quark model of Bicudo and Ribeiro [5]. The most systematic approach to such effects has been realised in the Nijmegen Unitarised Meson Model (NUMM), based on a manifestly unitary, coupled-channel Schrödinger equation, with kinematically relativistic corrections. The NUMM, in which all real and virtual two-meson channels containing mesons out of the ground-state pseudoscalar and vector nonets are coupled to the confined valence-quark sector via the mechanism, has been successfully applied to pseusoscalar and vector mesons [2] and also to the usually awkward scalar mesons [20,21]. Especially in the scalar case, the unitarisation contributions turn out to be so large that not only the original spectrum is completely distorted, but even the number of resonances is doubled [21]. This phenomenon allows e.g. to solve the long-standing problem of a light s meson needed in nuclear physics. Nevertheless, in view of the Dirac nature of quarks and the large velocities showing up in the equations for especially the light mesons, a relativistically covariant reformulation of the NUMM is desirable. This will also allow to describe the mechanism in a more fundamental way, and to include final-state interactions in the two-meson sector, based on t channel meson exchange. As a possible starting point, the relativistic constituent-quark model of Ref.[22] will be taken, but M reformulated in momentum space.

Work in collaboration with Prof. Eef van Beveren of the University of Coimbra.

The suppression of the J/y production in nucleus nucleus collision was proposed, more than ten years ago, by Matsui and Satz [23] as a signal for the formation of the quark-gluon plasma (QGP). The suggested mechanism is simple: when colour is liberated Debye screening prevents the creation of the bound state. Suppression of the J/y relative to Drell-Yan (DY) production was indeed observed by the NA38 collaboration in O-U and S-U collisions [24]. However, it was soon realized that conventional nuclear obsorption of the J/y should play, qualitatively at least, a similar rôle [25]. In recent years, a great deal of theoretical effort was developed in trying to clarify the origin of the J/y absorption that would be able to justify the observed suppression. In fact, a cross section of the order of 7mb, much larger than than the observed y N cross section, is needed. In this context, the rôle of interacting pre-resonant states in matter is to help achieve such large cross section [26]. More recently, the NA50 collaboration, from the analysis of the 1996 data, reported in addition to the anomalous suppression, a discontinuity or abrupt fall in the ratio J/y over DY around [27]. Based on a very simple and intuitive model [28] we were able to point out [28] that simple mechanisms of absorption in nuclear matter could be responsible for this behaviour. In fact, this simple test model was able to reproduce fairly well the data obtained by the NA38-50 experiments in a large range of energies and nuclei. We thus conclude that the only statement that can be made with certainty at present is that the drop in the ratio J/y over DY is due to absorption in a medium with increasing density and consequently with increasing opacity. Only future experiments, in particular ALICE at the LHC, can enable us to answer to the question of disentangling between J/y absorption and J/y destruction in a deconfined medium.

[1] P. Bicudo, L. Ferreira, C. Placido, and J. E. Ribeiro, Phys Rev C 56, (1997), 670-678; P. Bicudo and J.E. Ribeiro, Phys Rev C 55, (1997), 834-847.

[2] E. van Beveren, G. Rupp, T. A. Rijken, and C. Dullemond, Phys. Rev. D27, (1983).

[5] P. Bicudo and E. Ribeiro, Phys. Rev. D 42, 1611 (1990); D42, 1625 (1990); D42, 1635 (1990).

[7] N. Brambilla and A. Vairo, Phys. Lett. B 407, 167 (1997); Nucl. Phys. Proc. Suppl. B 64, 423 (1998).

[8] F. Karsch, in "QCD, 20 years later" eds. P. M. Zerwas and H. A. Kastrup (World Scientific, Singapore, 1993).

[9] C. D. Roberts and A. G. Williams, Prog. in Part. and Nucl. Phys. 33, 477 (1994).

[10] Particle Data Group, C. Caso et al. European Physical Journal C 3 (1998) 1; see

also http://pdg.lbl.gov/.

[11] N. Isgur and M. Wise, Phys. Lett. B 232, 113 (1989).

[13] E. J. Eichten, C. Hill and C. Quigg, Phys. Rev. Lett. D 71, 4116 (1993).

[14] W. Kwong and J. Rosner, Phys. Rev. D 44, 212 (1991) and refs. therein.

[15] M. Neubert, Phys. Rev. D 46, 1076 (1992); E. Bagan, P. Ball, V. M. Braun and H. G. Dosch, Phys. Lett. B 278, 457 (1992).

[17] H. G. Dosch, Phys. Lett. B190, 177 (1987);

[20] E. van Beveren, C.Dullemond, C.Metzger, J.E.Ribeiro, T.A.Rijken, and G.Rupp, Z. Phys. C30, 615 (1986).

[21] Eef van Beveren and George Rupp, Comment on "Understanding the scalar meson nonet", hep-ph/9806246, submitted for publication.

[22] P.C.Tiemeijer and J.A.Tjon, Phys. Lett. B277, 38 (1992).

[23] T. Matsui and H. Satz, Phys. Lett. B178 (1986) 416.

[24] C. Baglin et al., Phys. Lett. B 220 (1989) 471; Phys. Lett. B 251 (1990) 465; Phys. Lett. B 255 (1991) 459.

[25] Capella et al A. Capella et al., Phys. Lett. B 336 (1996) 316.

[26] G.T. Bodwin, E. Braaten and P.Lepage, Phys. Rev. D 51 (1995)1125; E. Braaten and S. Fleming, Phys. Rev. Lett. 74 (1995) 3327; Kharzeev and H. Satz, Phys. Lett. B 336 (1996) 316.

[27] M. Gonin Nuc. Phys. C610 (1996) 404; Nucl. Phys. C610 (1996) 552.

[28] J. Dias de Deus, C. Pajares and C.A. Salgado, Phys. Lett. B 407 (1997) 335; Phys. Lett. B 408 (1997) 417; J. Dias de Deus and J. Seixas, Phys. Lett. B430 (1998) 363.

Although QCD is the accepted fundamental theory of strong interactions, with quarks and gluons as its elementary degrees of freedom, at low and intermediate energies nucleons and mesons appear as the relevant degrees of freedom for the study of nuclei and nuclear reactions. The effective interactions between colourless hadrons, in particular the NN interaction, are determined through a combination of symmetry principles, reflecting the fundamental underlying microscopic theory, and phenomenology. In order to establish the limits of validity of this traditional picture, and to find out in what energy regime one has to go from the nucleon-meson picture to the quark-gluon picture, there is a need for extensions to higher energies of the commonly used NN interactions, calibrated so far only below the pion-production threshold. Naturally, in going to higher energies and above meson-production thresholds, a relativistically covariant and unitary formulation is required, both in the kinematics and the dynamics, in order to ensure a realistic description. Meson production in pp collisions, and other reactions involving light nuclei, allow us to explore the dynamics of nucleonic excitations N^* their decay couplings, and propagation in the nuclear medium in a contrôled way. On the other hand, employing Bethe-Salpeter and Bethe-Salpeter-Faddeev equations with covariant separable interactions makes a fully relativistic description possible of the binding energies and electromagnetic form factors of light nuclei as the deuteron and the triton.

In meson production reactions where there is no interplay of "doorway like" resonances and the impulse terms dominates the reaction mechanism, the measurement and calculation of cross sections puts constraints on theoretical quark models, through their predictions for meson-nucleon couplings. Ultimately it gives information on media effects on couplings and off-mass-shell-propagation.

In this project we propose to calculate the total cross section as well as the single and double differential cross sections for the reactions pp ® h pp, p ⁺d ® h pp and p ^-d ® h nn. We want to continue the program of investigating the importance of relativity in light nuclei, in particular for meson production dynamics [1]. The relativistic effects included will not be purely kinematical; they will accommodate for the Dirac structure for the S₁₁, D₁₃ and P₁₁ N^* propagation. Pauli-blocking and fermi-motion effects will be assessed. For the h production reactions mentioned, and contrarily to the p production case, we expect a suppression of the FSI, that makes the h production reactions ideal to test reaction mechanisms connected with the nucleonic resonances alone.

The cross section for the reaction near threshold is exceptionally sensitive to short-range exchange mechanisms in the two-nucleon system, because the main pion exchange term is ruled out by isospin conservation in the two-nucleon system. Pion production on a single nucleon under-predicts the empirical cross section by a large factor, and short-range exchange mechanisms dominate over the pion exchange amplitude given by chiral perturbation theory [2]. In this situation it appears natural to investigate the rôle of such other short-range mechanisms. The most obvious additional short-range mechanisms are those which involve transition couplings between different exchanged mesons, and those that involve excitation of intermediate virtual nucleon resonances by short-range exchange mechanisms. We then study the rôle of the low lying nucleon resonances beyond the D (1232) in the reaction near threshold. We consider the intermediate N(1440) (P₁₁) resonance is excited by the short-range scalar and vector meson exchanges, while the N(1535) (S₁₁) and N(1520) (D₁₃) resonances are excited by h and r meson exchange, respectively.

Recently, methods based on path-integral techniques allowed field-theory calculations for scalar particles, including ladder and crossed-ladder series [3]. When the status of the calculations reaches the point of dealing with Dirac particles and general couplings, the comparison of their results with the ones obtained with different Quasi-Potential formalisms will clarify the unsettled point of selecting between different covariant formulations for the description of nuclear systems. In the meantime, for the case of Dirac particles interacting through boson-exchange, we want to calculate the two-pion exchange contributions not included in the spectator-on-mass-shell formalism. Although a reasonable calibration of the NN models at low energies was already achieved within the spectator formalism [4], the restriction of the kernel to the lowest order truncation, affects its capability to be extended to higher energies.

The trinucleon bound-state system is still posing a serious problem to standard three-body descriptions with non-relativistic (NR) two-body nucleon-nucleon (NN) potentials only, almost invariably leading to an underbinding of the order of 0.5-1.0 MeV, besides persisting discrepancies in the electromagnetic form factors. As an alternative to the inclusion of explicit three-body forces, the Bethe-Salpeter-Faddeev (BSF) equation with realistic separable interactions has been employed to study the importance of relativity in the trinucleon [5] This way, effects from relativistic kinematics, Lorentz covariance, and fully off-shell intermediate nucleons are automatically included. However, so far the Dirac spin structure of the nucleons has been neglected. To derive covariant separable interactions - indispensable to obtain tractable equations – with the correct Dirac structure, the previously used hybrid method of covariantising separable expansions of realistic NR potentials cannot be used anymore. Therefore, we intend to develop a separable-approximation method for relativistic one-boson-exchange (OBE) interactions as of Ref.[6]. The proposed scheme is a four-dimensional generalisation of the NR Gamow Separable Approximation (GSA) method [7] which we favour because of the absence of any ambiguity that might conjure up spurious poles, even if, hypothetically, other methods could do with a somewhat lower rank of expansion. In a first phase, the NR GSA method must be developed in momentum space, making use of contour-rotation techniques. Next, we plan to apply the relativistic GSA to a phenomenological, purely scalar OBE interaction with attraction and repulsion. Finally, in a more distant phase of the project, an application to the fully relativistic OBE model of Ref. [2] is envisaged.

[1] J. Adam Jr, Alfred Stadler, M. T. Peña, Franz Gross, Phys. Lett.B 407, 97(1997).

[2] T.-S. H. Lee and D. O. Riska, Phys. Rev. Lett. 70, 2237 (1993).

[3] T. Nieuwenhuis, J. A. Tjon and Y. A. Simonov Few-Body Systems Supp. 7, 286 (1994); T. Nieuwenhuis and J. A. Tjon, Phys. Rev. Lett. 77, 814 (1996).

[4] A. Stadler and Franz Gross, Phys. Rev. Lett. 78, 26 (1997).

[5] G. Rupp and J. A. Tjon, Phys. Rev. C45, 2133 (1992).

[6] J. Fleischer and J. A. Tjon, Phys. Rev. D21, 87 (1980).

[7] M.Baldo, L.S.Ferreira, and L.Streit, Phys.Rev. C32, 685 (1985).

One of the most exciting subjects in contemporary nuclear physics is the search for the limits of stability of nuclei. They correspond to the position of the nucleon drip lines that delimit the stability valley. The knowledge of nucleon emission from ground state of spherical, as well as deformed nuclei, provide information on these limits. Recently, there has been an intensive experimental activity in measuring proton decay and a large variety of proton emitters were observed in the region of heavy nuclei with 50 < Z < 82, encompassing a large number of deformed systems, and defining almost completely the borders of proton stability. In this high Z region, the Coulomb barrier is sufficiently high to keep the outgoing proton close to the core nucleus long enough to be detected. Outside the charge range mentioned above, proton decay from a superdeformed state to a spherical one, was also observed very recently. All these measurements yield very important spectroscopic information on the nature of the decaying system, coined in the decay transition rates. Therefore, theoretical studies of there reactions are required for the understanding of the structure of these nuclei, and to determine their deformation. We plan to continue our reserch in this topic by aplying our exact model, developed in ref 1, and ref 2. to the latestes experimental results on proton emission. We want to focus also on the possibility of having more than one proton emitted.

[1] L. S. Ferreira, E. Maglione, R. Liotta, Phys. Rev. Lett. 78 (1997) 1640-1643;

[2] E. Maglione, L. S. Ferreira, R. Liotta, Phys. Rev. Lett. 81 (1998) 58;

The Equation of State (EOS) of nuclear matter at finite temperature is of great interest in the physics both of heavy ion collisions and of supernovae explosions. In recent experiments on heavy ion collisions at intermediate energies (1) the structure of light particle spectra observed, seems to indicate the presence of a first order phase transition. This can be interpreted as the liquid-gas phase transition, predicted for nuclear nuclear matter, taking into account finite size effects and Coulomb corrections. The nuclear EOS is also related to the maximal temperature a nucleus can sustain before reaching mechanical instability. This ``limiting temperature" is mainly the maximal temperature at which a compound nucleus can be observed. Its value can be inferred from fusion reactions between heavy ions at intermediate energies and tests the predictions derived from the microscopic EOS. Concerning the astrophysical relevance (2), we can mention the latest stage of the supernovae collapse, where the EOS of asymmetric nuclear matter at finite temperature plays a major rôle in determining the final evolution. The EOS is actually one of the less well known elements which enters in the collapse simulations.

We have developed in a previous work (3), the microscopic equation of state of the nuclear matter at finite temperature within the general Bloch and De Dominicis (BD) linked diagrammatic expansion. At the two-body correlation level the BD formalism contains terms which correspond to the Brueckner approximation, where the single particle occupation numbers are replaced by the finite temperature Fermi distributions. The remaining finite temperature diagrams vanish in the zero temperature limit and originate from the fact that at finite temperature any momentum state can be interpretated both as a particle and as a hole. The liquid gas phase transition of symmetric nuclear matter was identified, and the critical temperature extracted, using the Argonne v₁₄ as thebare NN interaction and a phenomenological three-body force adjusted to give the correct saturation point. The Argonne is a realistic NN interaction, but it is mainly a phenomelogical one. The determination of a three-body body force from general principles, is also an impossible task, and we are allways bound to use effective ones. Therefore, it is important to study the changes in the microscopic EOS when different NN realistic interactions, like for example the Bonn potential, and more sofisticated three-body forces are considered. We plan to develop these ideas in our future work and study not only nuclear matter at finite temperature, but also pure neutron and asymmetric matter, relevant to supernovae explosions.

[1] J. Bondorf et al., Phys. Rep. 257, 133 (1995) and refs. therein.

[3] M. Baldo and L. S. Ferreira, Phys. Rev. C, in press.

Till the mid 1990's the binding energy of the triton was undoubtly the only signature of the three-nucleon force. Recently, it became clear that the calculation of one observable -the nucleon vector analyzing power A_y (q ) for nd elastic scattering, below 3 MeV – with realistic nucleon-nucleon potentials presents a 30% discrepancy relatively to the experimental data, contrasting stunningly with the perfect description of the large variety of all the other polarization observables. The addition of the two-p exchange three-nucleon force, traditionally considered for the bound state calculation, produces no sizeable effect.

In a recent work [1], Huber and Friar argue that the A_y discrepancy must be seen as a signal of a three-nucleon force, of a form not contained in the two pion exchange based one always considered, and with a possibly new spin-orbit type of structure. We intend to explore Friar and Huber' suggestion, by investigating in the framework of a NN-N D coupled-channel calculation, the effect in the three-nucleon scattering observables of spin-orbit terms in the transition potentials mediated by vector-meson exchanges (r exchange), not considered till now.

[1] D. Huber and J.L.Friar , Phys. Rev. C, 58, 674, (1998);

We continue the work with Giorgio Valli, of University of Pavia, of studying compact minimal immersed submanifolds F: M® N of a real dimension 2n manifold M into a Kähler-Einstein manifold N, of complex dimension 2n. In a previous work we defined a natural complex structure Jw in an even dimension subbundle of TM and define pluriminimality with respect to this complex structure. Then we considered a continuous locally Lipschitz map , where cos q a are the Kähler angles of F. This map is smooth at certain open spaces and we compute D k giving an expression in terms of the Ricci tensor and the Kähler angles. We proved that if M is pluriminimal of 4 dimension, without complex points, and if N has negative first Chern class, then it is Lagrangian. The result was known for 2-dimensional M, due to J.G. Wolfson (1989). We will study higher dimension cases, for minimal and pluriminimal, as well the case of M to be of odd dimension 2n+1. We are starting with the simpliest case of M of dimension 3. This case is very important to understand all other higher dimension cases, including the even dimension cases, and we expect to solve this soon. We will use similar computations with a similar map k . A problem is the fact that k is not C² everywhere, and the complexity of its singularities increase with the dimension, and the minimal non-pluriminimal case.

This is a long project that we will continue to study and develop. It envolves knowledge of Hodge theory in Lipschitz manifolds developed by Sullivan and Teleman (on the 80s), axiomatic theory of Geodesic spaces of bounded curvature of Busemann, Alekxandrov (50s) and Nikolaev (80s), and axiomatic theory of harmonic spaces due to Deny, Brelot, Herve (60s). Recently J. Jost (1994-1997) has defined a new definition of harmonic map between metric spaces with target space of negative curvature (in the Busmann-Alekxandrov sense) and obtain some results similar to the smooth Riemannian case. We will study harmonic maps from an harmonic space into a geodesic metric space, studying different possible definitions, compare properties, and apply to the case of a Lipschitz manifold (that is both a geodesic and harmonic space). For example, we would like to know when the identity map of a space that is both geodesic and harmonic space, is an harmonic map. Moreover, we would like to obtain a sort of Liouvile type problems. In particular, we want to study harmonic maps between spaces with a polyhedral metric.