Topology of quantum gauge fields and duality (I): Yang-Mills-Higgs system in 2 + 1 dimensions

The basic role of the representation of the gauge group in characterizing the topological excitations of the vacuum is pointed out. For SU(N) gauge fields on a lattice, the topological excitations are monopoles in the adjoint representation of the dual group ¹SU(N). This leads to a dual representation of the Yang-Mills-Higgs system in 2 + 1 dimensions. For SU(3) the deal theory in a scalar theory with discrete Weyl symmetry S₃. In the presence of adjoint Higgs fields the Weyl symmetry is broken in the Higgs phase but restored by pseudo-particles in the confinement phase.     

Can the nearly conformal sextet gauge model hide the Higgs impostor?

New results are reported from large scale lattice simulations of a frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. We find that the chiral condensate and the mass spectrum of the sextet model are consistent with chiral symmetry breaking in the limit of vanishing fermion mass. In contrast, sextet fermion mass deformations of spectral properties are not consistent with leading conformal scaling behavior near the critical surface of a conformal theory. A recent paper could not resolve the conformal fixed point of the gauge coupling from the slowly walking scenario of a very small nearly vanishing β-function (DeGrand et al. [3]). It is argued that overall consistency with our new results is resolved if the sextet model is close to the conformal window, staying outside with a very small non-vanishing β-function. The model would exhibit then the simplest composite Higgs mechanism leaving open the possibility of a light scalar state with quantum numbers of the Higgs impostor. It would emerge as the pseudo-Goldstone dilaton state from spontaneous symmetry breaking of scale invariance. We will argue that even without association with the dilaton, the scalar Higgs-like state can be light very close to the conformal window. A new Higgs project of sextet lattice simulations is outlined to resolve these important questions.     

Diquark condensation in dense adjoint matter

We study SU(2) lattice gauge theory at non-zero chemical potential with one staggered quark flavor in the adjoint representation. In this model the fermion determinant, although real, can be both positive and negative. We have performed numerical simulations using both hybrid Monte Carlo and two-step multibosonic algorithms, the latter being capable of exploring sectors with either determinant sign. We find that the positive determinant sector behaves like a two-flavor theory, with the chiral condensate rotating into a two-flavor diquark condensate for , implying a superfluid ground state. Good agreement is found with analytical predictions made using chiral perturbation theory. In the ‘full’ model there is no sign of either onset of baryon density or diquark condensation for the range of chemical potentials we have considered. The impact of the sign problem has prevented us from exploring the true onset transition and the mode of diquark condensation, if any, for this model. Received: 28 September 2001 / Published online: 23 November 2001     

Lattice simulations for light nuclei: Chiral effective field theory at leading order

We discuss lattice simulations of light nuclei at leading order in the chiral effective field theory. Using lattice pion fields and auxiliary fields, we include the physics of instantaneous one-pion exchange and the leading-order S-wave contact interactions. We also consider higher-derivative contact interactions which adjust the S-wave scattering amplitude at higher momenta. By construction our lattice path integral is positive definite in the limit of exact Wigner SU(4) symmetry for any even number of nucleons. This SU(4) positivity and the approximate SU(4) symmetry of the low-energy interactions play an important role in suppressing sign and phase oscillations in Monte Carlo simulations. We assess the computational scaling of the lattice algorithm for light nuclei with up to eight nucleons and analyze in detail calculations of the deuteron, triton, and helium-4.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

Impossibility of adding gauge couplings to spontaneously broken N = 8 supergravity

When the four mass parameters of spontaneously broken N = 8 supergravity are taken to be equal, the theory possesses at global SU(4) symmetry. Since it contains massless vectors in the adjoint representation, it is tempting to add gauge interactions so as to make the SU(4) symmetry local. In this paper we show that it is impossible to do this in a way that is consistent with the spontaneous character of the symmetry breaking.     

Monopoles and holography

Several lattice collaborations performing simulations with 2+1 light dynamical quarks have experienced difficulties in fitting their data with standard N _f = 3 chiral expansions at next-to-leading order, yielding low values of the quark condensate and/ or the decay constant in the N _f = 3 chiral limit. A reordering of these expansions seems required to analyse these data in a consistent way. We discuss such a reordering, known as Resummed Chiral Perturbation Theory, in the case of pseudoscalar masses and decay constants, pion and kaon electromagnetic form factors and K _ℓ3 form factors. We show that it provides a good fit of the recent results of two lattice collaborations (PACS-CS and RBC/UKQCD). We describe the emerging picture for the pattern of chiral symmetry breaking, marked by a strong dependence of the observables on the strange quark mass and thus a significant difference between chiral symmetry breaking in the N _f = 2 and N _f =3 chiral limits. We discuss the consequences for the ratio of decay constants F _K /F _π and the K ℓ3 form factor at vanishing momentum transfer.

     

Anomalous discrete chiral symmetry in the Gross–Neveu model and loop gas simulations

We investigate the discrete chiral transformation of a Majorana fermion on a torus. Depending on the boundary conditions the integration measure can change sign. Taking this anomalous behavior into account we define a chiral order parameter as a ratio of partition functions with differing boundary conditions. Then the lattice realization of the Gross–Neveu model with Wilson fermions is simulated using the recent ‘worm’ technique on the loop gas or all-order hopping representation of the fermions. An algorithm is formulated that includes the Gross–Neveu interaction for N fermion species. The critical line m_c(g) is constructed for a range of couplings at N=6 and for N=2, the Thirring model, as examples.     

Numerical simulation tests with light dynamical quarks

Two degenerate flavors of quarks are simulated with small masses down to about one fifth of the strange quark mass by using the two-step multi-boson (TSMB) algorithm. The lattice size is with lattice spacing about which is not far from the N _t =4 thermodynamical cross-over line. Autocorrelations of different physical quantities are estimated as a function of the quark mass. The eigenvalue spectra of the Wilson-Dirac operator are investigated. Received: 20 June 2002 / Revised version: 30 August 2002 / Published online: 25 October 2002     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

Effective lagrangian of mesons and baryons in the strong-coupling expansion of lattice QCD

U(N) and SU(N) lattice QCD are considered. By using a method of the strong-coupling expansion, the effective lagrangian of hadrons is calculated up to the first order in 1/(g²N). For the Susskind lattice fermions, it is shown that chiral symmetry is spontaneously broken and as a result there appears the Nambu-Goldstone boson (pion). The fermion condensation 〈$\text{ψ}\text{ψ}$t>, the masses of hadrons and the pion decay constant are calculated and compared with the results of Monte Carlo (MC) simulations. In the strong-coupling region, our result of the order parameter 〈$\text{ψ}\text{ψ}$〉 coincides very well with that calculated by MC simulations.     

Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System

We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T ₁ through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T ₂ through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T ₁〉 ∝ Lln L and ln 〈 T ₂〉 ∝ L, where L is the system size. The actual value of T ₁ depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T ₂ is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity.     

Instantons and the U(1) problem

If instantons spontaneously break the chiral SU(N) × SU(N) symmetry of a non-abelian gauge theory, they break U(1) symmetry in a manner consistent with the chiral Ward identities of the theory. Excitations of the fermion vacuum play a crucial role in this process. A model calculation of the symmetry breaking effect shows a phenomenological structure which differs from that provided by models with many color degrees of freedom.     

Twelve massless flavors and three colors below the conformal window

We report new results for a frequently discussed gauge theory with twelve fermion flavors in the fundamental representation of the SU(3) color gauge group. The model, controversial with respect to its conformality, is important in non-perturbative studies searching for a viable composite Higgs mechanism beyond the Standard Model (BSM). In comparison with earlier work, our new simulations apply larger volumes and probe deeper in fermion and pion masses toward the chiral limit. Investigating the controversy, we subject the model to opposite hypotheses with respect to the conformal window. In the first hypothesis, below the conformal window, we test chiral symmetry breaking (χSB) with its Goldstone spectrum, Fπ, the χSB condensate, and several composite hadron states as analytic functions of the fermion mass when varied in a limited range with our best effort to control finite volume effects. In the second test, for the alternate hypothesis inside the conformal window, we probe conformal behavior driven by a single anomalous mass dimension under the assumption of unbroken chiral symmetry at vanishing fermion mass. Our results at fixed gauge coupling, based on the assumptions of the two hypotheses we define, show low level of confidence in the conformal scenario with leading order scaling analysis. Relaxing the important assumption of leading mass-deformed conformality with its conformal finite size scaling would require added theoretical understanding of the scaling violation terms in the conformal analysis and a comprehensive test of its effects on the confidence level of the fits. Results for the running coupling, based on the force between static sources, and preliminary indications for the finite temperature transition are also presented. Staggered lattice fermions with stout-suppressed taste breaking are used throughout the simulations.     

Numerical simulations of two-dimensional QED

We describe the computer simulation of two-dimensional QED on a 64 × 64 Euclidean space-time lattice using the Susskind lattice fermion action. The order parameter for chiral symmetry breaking and the low-lying meson masses are calculated for both the model with two continuum flavours, which arises naturally in this formulation, and the model with one continuum flavour obtained by including a nonsymmetric mass term and setting one fermion mass equal to the cut-off. Results are compared with those obtained using the quenched approximation, and with analytic predictions.     

Chiral symmetry breaking in lattice QCD by the green function method

The chiral symmetry breaking at strong coupling in lattice QCD for naive and Wilson fermion cases is investigated by the Green function method with the Hartre-Fock approximation. Solving the coupled equations of the Green functions for the composite fields we compute the pion mass which directly satisfies the current algebra mass formula for both fermion cases.     

AnSU(2)l ⊗SU(2)r symmetric Yukawa model in the symmetric phase

The lattice regularizedSU(2)l ?SU(2)r symmetric scalar fermion model with explicit mirror fermions is investigated in the phase with unbroken symmetry. In the present work numerical Monte Carlo calculations with dynamical fermions are performed on 4³·8 and 4³·16 lattices near the expected perturbative Gaussian fixed point. The bare Yukawa coupling of the mirror fermion is fixed at zero. Global symmetries of the model are discussed, and the numerical results are supported by lattice perturbation theory.     

A Wilson-Yukawa Model with undoubled chiral fermions in 2D

We consider the fermion mass spectrum in the strong coupling vortex phase (VXS) of a lattice fermion-scalar model with a global U(1)_L × U(1)_R, in two dimensions, in the context of a recently proposed two-cutoff lattice formulation. The fermion doublers are made massive by a strong Wilson-Yukawa coupling, but in contrast with the standard formulation of these type of models, in which the light fermion spectrum was found to be vector-like, we find massless fermions with chiral quantum numbers at finite lattice spacing. When the global symmetry is gauged, this model is expected to give rise to a lattice chiral gauge theory.     

Flavor symmetry breaking and scaling for improved staggered actions in quenched QCD

We present a study of the flavor symmetry breaking in the pion spectrum for the p4-improved fermion action. Three different variants of the p4 action – p4fat3, p4fat7, and p4fat7tad – are compared to the Asqtad and naive staggered actions. To study the pattern of symmetry breaking, we measure all 15 pion masses in the four-flavor staggered theory. The measurements are done on a quenched gauge background, generated using a one-loop improved Symanzik action with β=10/g²=7.40, 7.75, and 8.00, corresponding to lattice spacings of approximately a= 0.31 fm, 0.21 fm, and 0.14 fm. PACS 11.15.Ha; 11.30.Rd; 12.38.Aw; 12.38.-t; 12.38.Gc     

Fermion Determinant with Dynamical Chiral Symmetry Breaking

One-loop fermion determinant is discussed for the case in which the dynamical chiral symmetry breaking caused by momentum-dependent fermion self-energy Σ(p²) takes place. The obtained series generalizes the heat kernel expansion for hard fermion mass.