Paulo A. Faria da Veiga

Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics

We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell’Mann-Ne’eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3)c gauge and global SU(3)f flavor symmetries. We take the hopping parameter κ and the pure gauge coupling β satisfying the strong coupling regime condition 0⩽β⪡κ⪡1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3)f symmetry, the 56 baryon states admit a decomposition into 8×2 states associated with a spin 1∕2 octet and 10×4 states associated with a spin 3∕2 decuplet. The states are labeled by t...

Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics

We show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quark-fluon dynamics. We also give a proof of confinement below the two-meson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCD model with Wilson action, SU(3)f global and SU(3)c local symmetries. We work in the strong coupling regime, such that the hopping parameter κ>0 is small and much larger than the plaquette coupling β>1/g02⩾0 (β⪡κ⪡1). In the quantum mechanical physical Hilbert space H, a Feynman-Kac type representation for the two-meson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the two-meson truncated correlation and the energy-momentum spectrum of the model. The total spin operator J and its z-component Jz are defined by using π∕2 rotations about t...

ON THE ABSENCE OF PENTAQUARK STATES FROM DYNAMICS IN STRONGLY COUPLED LATTICE QCD

We consider an imaginary time functional integral formulation of a two-flavor, 3+1 lattice QCD model with Wilsons action and in the strong coupling regime (with a small hopping parameter, κ > 0, and a much smaller plaquette coupling, , so that the quarks and glueballs are heavy). The model has local SU(3)c gauge and global SU(2)f flavor symmetries, and incorporates the corresponding part of the eightfold way particles: baryons (mesons) of asymptotic mass ≈-3 ln κ(≈-2 ln κ). We search for pentaquark states as meson–baryon bound states in the energy–momentum spectrum of the model, using a lattice Bethe–Salpeter equation. This equation is solved within a ladder approximation, given by the lowest nonvanishing order in κ and β of the Bethe–Salpeter kernel. It includes order κ2 contributions with a exchange potential together with a contribution that is a local-in-space, energy-dependent potential. The attractive or repulsive nature of the exchange interaction depends on the spin of the meson–baryon states. The Bethe–Salpeter equation presents integrable singularities, forcing the couplings to be above a threshold value for the meson and the baryon to bind in a pentaquark. We analyzed all the total isospin sectors, I = 1/2, 3/2, 5/2, for the system. For all I, the net attraction resulting from the two sources of interaction is not strong enough for the meson and the baryon to bind. Thus, within our approximation, these pentaquark states are not present up to near the free meson–baryon energy threshold of ≈-5 ln κ. This result is to be contrasted with the spinless case for which our method detects meson–baryon bound states, as well as for Yukawa effective baryon and meson field models. A physical interpretation of our results emerges from an approximate correspondence between meson–baryon bound states and negative energy states of a one-particle lattice Schrodinger Hamiltonian.

EXACT DYNAMICAL EIGHTFOLD WAY BARYON SPECTRUM AND CONFINEMENT IN STRONGLY COUPLED LATTICE QCD

We obtain from the quark–gluon dynamics the eightfold way baryon spectrum exactly in an imaginary time functional integral formulation of 3+1 lattice QCD with Wilsons action in the strong coupling regime (small hopping parameter 0 < κ ≪ 1 and much smaller plaquette coupling ). The model has SU(3)c local gauge and global SU(3)f flavor symmetries. A decoupling of the hyperplane method naturally unveils the form of the baryon composite fields. In the subspace of the physical Hilbert space of vectors with an odd number of quarks, the baryons are associated with isolated dispersion curves in the energy–momentum spectrum. Spectral representations are derived for the two-baryon correlations, which allow us to detect the energy–momentum spectrum and particles as complex momentum space singularities. The spin 1/2 octet and spin 3/2 decuplet baryons have asymptotic mass -3ln κ and for each baryon there is an antibaryon with identical spectral properties. An auxiliary function method is used to obtain convergent expansions for the masses after subtracting the singular part -3ln κ. The nonsingular part of the mass is analytic in κ and β, i.e. the expansions are controlled to all orders. For β = 0, all the masses have the form M = -3ln κ - 3κ3/4 + κ6r(κ), with r(κ) real analytic. Although we have no Lorentz symmetry in our lattice model, we show that there is a partial restoration of the continuous rotational symmetry at zero spatial momentum, which implies that for all members of the octet (decuplet) r(κ) is the same. So, there is no mass splitting within the octet and within the decuplet. However, there is an octet–decuplet mass difference of at β = 0; the splitting persists for β ≠ 0. We also obtain the (anti)baryon dispersion curves which admit the representation , where and is of . For the octet, is jointly analytic in κ and in each pj, for small . A new local symmetry, which we call spin flip, is used to establish constraints for the matrix-valued two-baryon correlation and show that all the octet dispersion curves are the same and that the four decuplet dispersion curves are pairwise-identical and depend only on the modulus of the spin z-component. Using a correlation subtraction method we show that the spectrum generated by the baryon and antibaryon fields is the only spectrum, in the odd quark subspace of physical states, up to near the baryon–meson threshold of ≈ -5ln κ. Combining this result with a similar result for the mesons, with mass ≈ -2ln κ, shows that the only spectrum in the entire space of states, up to near the two-meson threshold of ≈ -4ln κ, is generated by the eightfold way hadrons. Hence, for 0 < κ ≪ β ≪ 1, we have shown confinement up to near this threshold.

One-baryon spectrum and analytical properties of one-baryon dispersion curves in 3 + 1 dimensional strongly coupled lattice QCD with three flavors

Considering a 3 + 1 dimensional lattice quantum chromodynamics (QCD) model defined with the improved Wilson action, three flavors, and 4 × 4 Dirac spin matrices, in the strong coupling regime, we reanalyze the question of the existence of the eightfold way baryons and complete our previous work where the existence of isospin octet baryons was rigorously solved. Here, we show the existence of isospin decuplet baryons which are associated with isolated dispersion curves in the subspace of the underlying quantum mechanical Hilbert space with vectors constructed with an odd number of fermion and antifermion basic quark and antiquark fields. Moreover, smoothness properties for these curves are obtained. The present work deals with a case for which the traditional method to solve the implicit equation for the dispersion curves, based on the use of the analytic implicit function theorem, cannot be applied. We do not have only one but two solutions for each one-baryon decuplet sector with fixed spin third compone...

Scaled lattice fermion fields, stability bounds, and regularity

We consider locally gauge-invariant lattice quantum field theory models with locally scaled Wilson-Fermi fields in d = 1, 2, 3, 4 spacetime dimensions. The use of scaled fermions preserves Osterwalder-Seiler positivity and the spectral content of the models (the decay rates of correlations are unchanged in the infinite lattice). In addition, it also results in less singular, more regular behavior in the continuum limit. Precisely, we treat general fermionic gauge and purely fermionic lattice models in an imaginary-time functional integral formulation. Starting with a hypercubic finite lattice Λ⊂(aZ)d, a ∈ (0, 1], and considering the partition function of non-Abelian and Abelian gauge models (the free fermion case is included) neglecting the pure gauge interactions, we obtain stability bounds uniformly in the lattice spacing a ∈ (0, 1]. These bounds imply, at least in the subsequential sense, the existence of the thermodynamic (Λ ↗ (aZ)d) and the continuum (a ↘ 0) limits. Specializing to the U(1) gauge gro...

ISOSPIN–SPIN INTERCHANGE SYMMETRY AND TWO-BARYON BOUND STATES IN (3+1)-DIMENSIONAL LATTICE QCD WITH TWO-FLAVORS AND STRONG COUPLING

We determine two-baryon bound states in a 3+1 lattice QCD model with improved Wilson action and two flavors. We work in the strong coupling regime: small hopping parameter κ > 0 and much smaller plaquette coupling β > 0. In this regime, it is known that the low-lying energy–momentum spectrum is comprised of baryons and mesons with asymptotic masses -3 ln κ and -2 ln κ, respectively. We show that the dominant baryon–baryon interaction is an order κ2 space-range-one -exchange potential. We also show that this interaction has an important and novel isospin–spin interchange symmetry relating the various possible bound states, and then governing the two-baryon spectral structure. Letting S(I) denote the total spin (total isospin) of the two-baryon bound states, S, I = 0, 1, 2, 3, we find bound states with asymptotic binding energy κ2/4, for I+S = 1, 3, and 4 (here, with I = S = 2); κ2/12, for I+S = 0, 2, 4 and 3 (here, with I = 1, 2). In particular, we show that the two-baryon spectrum contains deuteron (I = 0), diproton (I = 1) and dineutron (I = 1)-like bound states. Using the isospin–spin symmetry, we can circumvent the lack of spin symmetry of the lattice action and show they all have the same asymptotic binding energy, namely κ2/4. Our analysis uses convenient two and four-baryon correlations, their spectral representations and a lattice Bethe–Salpeter equation, which is solved in a ladder approximation. For the isospin, spin part of the interaction, we obtain a permanent representation which describes the interaction of the individual spins and isospins of the quarks of one baryon with those of the other baryon.

Hadron‐Hadron Bound States From First Principles

We present a summary of our results on the hadron spectrum and hadron‐hadron bound state spectrum which were obtained as part of a program which tries to bridge the gap between QCD and Nuclear Physics. We analyzed SU(3) lattice QCD models in the strong coupling regime (small hopping parameter κ > 0 and large glueball mass). We considered an imaginary time formulation for 2 + 1 and 3 + 1 dimensions models with one and two flavors. For the more algebraically complex and more realistic model in 3 + 1 dimensions, 4 × 4 spin matrices, and two flavors, we show the existence of three‐quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses −3ln κ and isolated dispersion curves. Baryon‐baryon bound states of isospin zero are found with binding energy of order κ2. The two‐particle analysis is performed using a ladder approximation to a lattice Bethe‐Salpeter equation written with the help of relative coordinates adapted to the lattice. The dominant baryon‐baryon interaction is an energy‐independent spatial range‐one attractive potential with an O(κ2) strength. There is also attraction arising from gauge field correlations associated with six overlapping bonds, but it is counterbalanced by Pauli repulsion to give a vanishing zero‐range potential. The overall range‐one potential results from a quark, antiquark exchange with no meson exchange interpretation since the spin indices are not of a meson particle; we call it a quasimeson exchange. The repulsive or attractive nature of the interaction depends on the isospin and spin of the two‐baryon state.We present a summary of our results on the hadron spectrum and hadron‐hadron bound state spectrum which were obtained as part of a program which tries to bridge the gap between QCD and Nuclear Physics. We analyzed SU(3) lattice QCD models in the strong coupling regime (small hopping parameter κ > 0 and large glueball mass). We considered an imaginary time formulation for 2 + 1 and 3 + 1 dimensions models with one and two flavors. For the more algebraically complex and more realistic model in 3 + 1 dimensions, 4 × 4 spin matrices, and two flavors, we show the existence of three‐quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses −3ln κ and isolated dispersion curves. Baryon‐baryon bound states of isospin zero are found with binding energy of order κ2. The two‐particle analysis is performed using a ladder approximation to a lattice Bethe‐Salpeter equation written with the help of relative coordinates adapted to the lattice. The dominant baryon‐b...

Baryon-baryon bound states in strongly coupled lattice QCD

We determine baryon-baryon bound states in 3+1 dimensional SU(3) lattice QCD with two flavors, 4x4 spin matrices, and in an imaginary-time functional integral formulation. For small hopping parameter, {kappa}>0, and large glueball mass (strong coupling regime), we show the existence of three-quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses -3ln{kappa} and isolated dispersion curves. We only consider the existence of bound states of total isospin I=0,3. Using a ladder approximation to a lattice Bethe-Salpeter equation, baryon-baryon bound states are found in these two sectors, with asymptotic masses -6ln{kappa} and binding energies of order {kappa}{sup 2}. The dominant baryon-baryon interaction is an energy-independent spatial range-one potential with an O({kappa}{sup 2}) strength. There is also attraction arising from gauge field correlations associated with six overlapping bonds, but it is counterbalanced by Pauli repulsion to give a vanishing zero-range potential. The overall range-one potential results from a quark, antiquark exchange with no meson-exchange interpretation; the repulsive or attractive nature of the interaction does depend on the isospin and spin of the two-baryon state.

Nonlocal interactions and the excitation spectrum in lattice quantum scalar field models

We determine the effects of nonlocal, nonlinear interactions on the excitation spectrum of lattice quantum field scalar models. We consider perturbations of a quantized discrete string formally self-adjoint Hamiltonian operator on the lattice d, and with a large mass coefficient for the quadratic term. The low-lying energy–momentum spectrum has an isolated dispersion curve and a two-particle (first) band. We analyse a ladder approximation of the Bethe–Salpeter equation on the lattice, for a weak perturbation of the type ∑d [λ6 : ()6 : +V(()], λ6 > 0, and consider the spectral interval starting at zero and extending to near the three-particle threshold. For space dimension d = 1,2 and V(()) = λ1 : ()4 :, we find that a bound state occurs either below (if λ1 0), but not both. This agrees with recent results where bound states were obtained for the stochastic dynamics generator associated with the relaxation rate to equilibrium in weakly coupled stochastic Ginzburg–Landau models with continuous time and on a spatial lattice d. These results are in contrast, however, with those obtained for V(()) = λ2 : ()3(− Δ)() :. For this case, surprisingly, we show that stable particles exist simultaneously above and below the band, for d = 1,2, regardless of the sign of the coupling λ2. If V(()) = λ3 : ()2(− Δ2)() :, the ladder analysis is inconclusive. If d = 3, 4, ..., no bound states exist in the spectral region we consider.