Effective lagrangian and dynamical symmetry breaking in strongly coupled lattice QCD

A method is proposed for computing effective lagrangians in QCD with N colors using lattice regularization. The meson field lagrangian is worked out in detail in the strong coupling limit with various lattice fermion formulations. For generalized Susskind fermions the spontaneous breakdown U(n) ? U(n) → U(n) (diagonal) is found at large N and a generalized version of the non-linear σ model emerges in a natural way. The Nambu-Goldstone spectrum is investigated and a continuous transition is made to Wilson fermions, for which the effective potential and the ππ scattering amplitude are tested on chiral symmetry. Large d (=dimension) approximations are compared with the large N limit and applied to N = 3.     

Mass gap and universality for euclidean O(N) and CPN−1 models

From a suitably defined correlation function we evaluate the strong coupling expansion for the mass gap of an euclidean version of the O(N) models in 2D. Good agreement is found for N = 0, 1 and 2 with the known values of the critical temperature and for N ? 3 with the continuum mass gap as evaluated in an hamiltonian approach. Another test of universality based on the use of an asymmetric lattice also yields good results. An analogous discussion for the CP^N?1 models is performed.     

Comparing O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions to SU(N) gauge theories in 3 + 1 dimensions

We have computed the scale breaking Λ parameters of the euclidean and hamiltonian formulations of the lattice regulated O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions in terms of the Λ_PV parameters of the Pauli-Villars regulated continuum models. Using lattice perturbation theory, the renormalized mass gap has been determined in terms of Λ_PV for each model. These results are compared to analogous calculations in SU(N) gauge theories.     

From random to self-avoiding walks

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π _i=0 ^N?2 Π_j=i+2/N(1?ωδ_ij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.     

The transition from the lattice to the continuum: CPN−1 models at large N

We study in detail the large N solution for the CP^N?1 models on a euclidean lattice and their phase structure in D dimensions. In the two-dimensional case, we evaluate some physical quantities and discuss their behaviour in the continuum limit. In particular, for the internal energy, we found the analytic expression for the perturbative tail which, at large β, obscures the predicted renormalization group behaviour; such a tail, which, as we show, is reproduced by a mean field calculation at N=∞, must be subtracted in order to define a suitable scaling quantity. We also investigated the behaviour of the topological charge and we found no perturbative tail in agreement with previous Lüscher's results. Finally, some aspects of the supersymmetric case are discussed.     

On the lattice approach to the maximum Higgs boson mass

If the triviality upper bound on the Higgs boson mass m_H occurs for strong self-coupling, inferring properties of the Higgs from the euclidean propagator is in principle theoretically difficult whether in coordinate or momentum space. In that case, common methods of identifying m_H in lattice field theory simulations may produce a value for which is at best distantly related to the true upper limit. We discuss some shortcomings and ambiguities of recent results suggesting that the maximum occurs for weak coupling and emphasize potential complications due to finite-size and non-Lorentz-invariant effects of the lattice. The situation is illustrated by reference to the behavior in an analytically soluble approximation based on a 1/N expansion.     

Hamiltonian for the twisted reduced model of large-N QCD

We propose a Hamiltonian for the reduced model with twist of theSU(N→∞) lattice gauge theory. Also we discuss how information as the mass gap, say, might survive under reduction and consider the euclidean Schwinger-Dyson equations for multiloop expectation values.     

Towards the phase structure of euclidean lattice gauge theories with fermions

We propose the phase structure of abelian and non-abelian lattice gauge theories with fermions. We especially analyse Wilson's lattice action with euclidean discrete space-time. We mainly analyse $\text{〈}\text{ψ}{}_{\mathrm{n}}\text{ψ}{}_{\mathrm{n}}\text{〉}$ as an order parameter for the fermion-gauge coupled system. The Wilson loop integral and plaquette-plaquette two-point function are also useful in working out abelian phase diagrams. We will discuss physical implications of the phase diagrams, especially for the mass spectrum in the lattice continuum limit and chiral symmetry breaking. The 1/N expansion and a random walk idea are used in the formulation and play an important role in computing meson and baryon propagators in the strong coupling limit.     

Potentials in pure QCD on 324 lattices

A range of properties of pure SU (3) gauge theory are studied numerically on euclidean lattices of 32⁴ points at β values of 6.29, 6.585 and 6.88. Potentials between static pairs of colour charges in the triplet, octet and sextet representations are calculated. The lowest energy (the A_1g) gluonic modes are considered for each of these plus the E_u mode for the triplet case. In the calculations, the use of extended operators on the lattice is examined. The results are assessed for scaling but do not agree adequately with asymptotic behaviour based on the two-loop β-function.     

The large-N deconfinement phase transition in a partially reduced Eguchi-Kawal model

We construct generalized twisted Eguchi-Kawai models which for large-N reduce space-time to a lattice of arbitrary size. Large-N lattice gauge theory at finite temperature is investigated in a model on a lattice with L₀ time slices and two lattice points in very time slice. We observe the large-N deconfinement phase transition in the weak coupling region. Assuming asymptotic scaling we find a transition temperature T_c = (101±4)Λ_L.     

On the renormalization of the quantum field theory of point-like monopoles and charges

The quantum field theory of point-like monopoles and charges is first formulated on a euclidean lattice for a convenient regularization. The regularization preserves the peculiar features of the theory, namely those related to the invariance and to the quantization condition. The partition function is expressed as a path integral over the particle's closed paths and the action is constructed in terms of arbitrary surfaces having those paths as boundaries. The possible divergences of the continuum limit are discussed, in particular the vacuum polarization ones. It is found that, although both the electric charge Q and the magnetic charge G are renormalized as Q = Z_QQ_R and G = Z_GG_R, the quantization condition is preserved by the renormalization i.e. Z_QZ_G = 1 so that QG = Q_RG_R = 2πn. Due to the dual symmetry of the theory, then, for Q = G we get Z_Q = Z_G = 1.     

Reduced Dirac-Kähler lattice fermion action

It is shown that the lattice Dirac-Kähler action is reducible under a chiral-like transformation. This provides a new lattice fermion action for spinors that have 2^d−1 components (instead of 2^d), with the property that, in the free case, each component satisfies the lattice euclidean Klein-Gordon equation. Reflection positivity is satisfied on the lattice, thus assuring a (positive) physical Hilbert space. In d = 4 dimensions the spinors have 8 components, and the correct physical chiral anomaly in the continuum limit. The action is suitable for QCD quarks which, in the continuum limit, are described by Dirac spinors that occur in flavor doublets.     

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

The finite-element approach to lattice field theory is both highly accurate (relative errors 1/N ², whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p ²/2+q ^2k /2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.     

Simulation of euclidean quantum field theories by a random walk process

A new Monte Carlo method for euclidean lattice field theory is introduced by writing the Boltzmann distribution e^?s as a solution of a diffusion type equation and constructing the associated random walk process. It is practically tested for a quantum mechanical model and a non-compact version of lattice QCD. It is explained where the main interest in this algorithm lies: the diffusion process coming from an action that can be generalized to include non-conservative forces. This possibility is exploited in our QCD version to implement gauge fixing without Faddeev-Popov ghosts.     

Approximate formula for second-order rayleigh scattering from N acoustical scatterers

A model of second-order multiple scattering from N Rayleigh acoustical scatterers distribted at random throughout a volume V is analyzed, physically motivated approximations being used. The result is a simple expression for the angular distribution of the total second-order Rayleigh scattered flux that is analogous to the usual formula for first-order scattering from N independent scatterers.     

1/N series for quantum anharmonic oscillator eigenvalues and green functions

The large orders of the 1/N series for the O(N)-symmetric anharmonic oscillator are computed here in the functional approach. We explicitly give the Kth order of the 1/N series up to O(K^?1) for all energy eigenvalues and for the two-point Green function. Our results are derived considering solutions in a finite (but large) time interval (euclidean time).     

Quenched master fields

We derive an exact algebraic (master) equation for the euclidean master field of any large-N matrix theory, including quantum chromodynamics. The master equation is the quenched Langevin equation. The master field, a translationally covariant function of (uniform) random momenta and (gaussian) random noise, is easily constructed in perturbation theory.     

Field-theoretical treatment of approximations to radiative transitions in atoms

Starting with a field-theoretical many-body definition of eigenvalues and radiative transition matrix elements for atomic systems, a systematic approach is taken to approximations of the exact results. The guiding principle is the maintenance of gauge invariance (GI) in radiative transition S matrix elements. At the level of the one Coulomb exchange approximation in both the one-electron and the electron-hole propagator kernels, one obtains the well-known Hartree-Fock (HF) and random phase approximations (RPA). A detailed discussion and comparison of various approaches to RPA is made, in the case of both N and N ? 1 electron shielding (the regular HF and HF with frozen relaxed core—FRC). In the former case, a new and considerably simpler form of the RPA equations are obtained than heretofore proposed equivalent forms. Finally, a different approximation than the usual HF and RPA, involving higher-order correlations, is developed to illustrate how such approximations can be systematically generated.     

Low energy QCD predictions from non perturbative methods

We show that by using lattice results about the euclidean ?-propagator, it is possible to give theoretical predictions on the low energy behavior of thee ⁺ e ^?→had. cross section. Furthermore, we present a comparison of perturbative QCD results (plus certain sum rules corrections) with non perturbative results from the lattice in the intermediate energy region.     

Vacuum background fields in QCD as a source of confinement

Quarks and gluons in the vacuum background field with a finite correlation length d are shown to be linearly confined inside white states. The string tension for an arbitrary representation of SU(N) is obtained in terms of d and the gluon condensate and agrees with numerical data and large N behaviour. The QCD string picture is shown to emerge asymptotically at large euclidean distances.