Monte Carlo simulation of SU(2) gauge theory with fermions on a four-dimensional lattice

After integration over the fermions in an SU(2) lattice gauge theory, the effective fermionic action may be expressed as a sum over all possible closed gauge field loops with corresponding weight factors. We approximate this sum and perform a Monte Carlo simulation of a coupled fermion-gauge system on a 4⁴ lattice. We compare our results for 〈S_eff〉 and $\text{〈}\text{ψ}\text{ψ〉}$ for different values of the gauge field coupling β and fermion coupling κ with the free fermion theory on a lattice. 〈S_eff〉 turns out to be quite small for $\text{κ?}\text{1}\text{8}$.     

Monte Carlo calculations with fermions: The Schwinger model

An algorithm for Monte Carlo simulation of lattice gauge theories with fermions is presented. The method is applied to the Schwinger model with two flavors of massless fermions, formulated on a two-dimensional euclidean lattice. Preliminary results of the Monte Carlo iteration of this system are presented, with special emphasis on the behavior of the Wilson loop and bilocal chiral correlation functions such as $\text{〈}\text{ψ}\text{(1 + γ}{}_5\text{)ψ(x)}\text{ψ}\text{(1 ? γ}{}_5\text{)ψ(y)〉}$.     

Monte Carlo calculation of the thermodynamic properties of a one-dimensional fermion lattice model

The generalized Trotter formula is used to derive two different classical representations of the partition function of a one-dimensional fermion model. Shortchain calculations are used to study the corresponding approximants for the energy and specific heat. A Monte Carlo technique has been used to calculate the temperature-dependent properties of a chain of 64 sites.     

Chiral phase transition in lattice QCD with dynamical quarks

The effect of virtual light quark loops on the chiral phase transition in lattice QCD with staggered fermions is investigated by employing the pseudo-fermion Monte Carlo method on a 6³ × 2 lattice. The variation in the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ is found to become less sharp than the quenched case, indicating a second order chiral phase transitioon in the full theory.     

The high-temperature behaviour of lattice QCD with fermions

By Monte Carlo simulation on the lattice, we calculate the high-temperature behaviour of the energy density ? in SU(2) and SU(3) QCD with Wilson fermions. From the leading term of the hopping parameter expansion, we find that ? converges very rapidly to the Stefan-Boltzmann limit of an asymptotically free quark-gluon gas. The behaviour in the non-asymptotic region indicates that chiral symmetry restoration occurs at a higher temperature (T_ch) than colour deconfinement (T_c), with $\text{T}{}_{\mathrm{<mtext>ch</mtext>}}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{≈ 1.3}$.     

Monte Carlo estimates of the SU(2) mass gap

Combining Monte Carlo and variational techniques, we compute the mass gap (or glueball mass) in the SU(2) lattice gauge theory. We obtain the estimate $\text{m}{}_{\mathrm{g}}\text{=(2.4 ± 0.6)}\text{K}\text{, K}$ being the string tension. We also discuss previous results derived by numerical or strong coupling methods and present Monte Carlo data on the second moment of the correlation length.     

A method for estimating glueball-meson mixing in lattice QCD

A systematic expansion of lattice QCD amplitudes based on the replica trick is discussed, the leading term of which is the quenched approximation. A parameter is defined that estimates the mixing between glueball and q$\text{q}$ meson states and provides a test for the reliability of the quenched approximation. The procedure is illustrated by an explicit Monte Carlo calculation for a model system on a one-dimensional lattice.     

Λ-parameters for asymmetric lattice gauge theories with fermions

Dimensional quantities obtained from Monte Carlo simulations on the lattice depend on the lattice mass parameter, Λ_L. To make a connection with continuum physics, a relationship is needed between Λ_L and the Λ-parameters of the continuum theory. This has been done for the euclidean symmetric lattice by others. However, in order to incorporate finite temperature into Monte Carlo studies, or to study the transition from the euclidean formulation to the hamiltonian formulation of gauge theories, asymmetric lattices (a_s≠a_t) may be used. In this paper, the assymetric calculations are extended and the ratio $\text{Λ}{}_{\mathrm{<mtext>min</mtext>}}\text{Λ}{}_{\mathrm{<mtext>L</mtext>}}$, where Λ_min is the continuum mass parameter in the minimal subtraction scheme, is given to one loop for n_f flavors of Wilson and Susskind massless fermions on an asymmetric four-dimensional lattice for two different asymmetric lattice actions.     

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.     

Massless composites with massive constituents

We construct model field theories in which a confining gauge interaction binds massive elementary fermions into massless composite particles. The massless composites are either Goldstone bosons or $\text{spin}\text{-}\text{1}\text{2}$ fermions. In these models, the manner in which exact chiral symmetries are realized changes at a critical value of the elementary fermion mass of order (e²/16π²)Λ, where Λ is the confinement scale and e is a weak gauge coupling.     

On the universality class of the deconfinement transition in lattice gauge theories at finite temperature

We study the (2 + 1)-dimensional Z(2) lattice gauge theory at a finite temperature by Monte Carlo simulation with system size up to three million variables. Our data indicate that the critical exponent β of the Polyakov line correlation function is greater than $\text{1}\text{8}$, in contradiction to a recent conjecture based on renormalization group arguments.     

General correction theorems on decomposition formulae of exponential operators and extrapolation methods for quantum Monte Carlo simulations

It is proved that the trace of the generalized Trotter formula $\text{Z(n) ≡ Tr[II exp(}\text{A}{}_{\mathrm{j}}\text{n]}{}^{\mathrm{n}}$ is an even function of n, when all A_j are symmetric, namely A^t_j = A_j, togethern with some generalizations. This yields a new extrapolatio n method of the form $\text{Z(n) ? Z(∞) + a/(n}{}^2\text{+ b)}$ for large n in quanntum Monte Carlo simulations.     

Studies of chiral symmetry breaking in SU(2) lattice gauge theory

We study chiral symmetry breaking (_χSB) in SU(2) lattice gauge theory with quarks in the $\text{l =}\text{1}\text{2}\text{, l = 1, l =}\text{3}\text{2}$, and l = 2 representations of the color group. We perform Monte Carlo evaluations of $\text{〈}\text{ψ}\text{ψ〉}$ in the quenched approximation and extract the relevant length scales for _χSB. We revise a previous estimate for the ratio between the chiral symmetry restoration temperatures for fundamental and adjoint quarks and obtain $\text{T}{}_{\mathrm{l\; =\; 1}}\text{/T}{}_{\mathrm{l}}\text{=}\text{1}\text{2}\text{～ 8}$. Our results for the higher representations, $\text{l =}\text{3}\text{2}\text{and}\text{l = 2}$, are consistent with Casimir scaling and give C₂g_mom² ～ 4. Many aspects of our calculational method are explained in detail. The issues discussed include the relation between _χSB in the quenched approximation and the spectrum of the Dirac operator, the flavor symmetries of euclidean staggered fermions, estimates of finite-size effects and the reliability of m → 0 extrapolations on finite lattices.     

Flavour and superunification

We consider the possibility of a non-trivial embedding of $\text{(}\text{10}\text{+}\text{5}\text{)}\text{SU}\text{(5)}$ families of spin if$\text{1}\text{2}$ left-handed fermions in a combination of irreducible massless supermultiplets of N extended supersymmetry. We demand the whole spectrum of spin $\text{1}\text{2}$ states to be anomaly free with respect to SU(N). This turns out to be a necessary condition for the absence of anomalies at the SU(5) level. We find two classes of models, with spin $\text{1}\text{2}$ fermions in SU(N) representations associated to one- and two-column Young tableaux, respectively, in which each irreducible massless multiplet occurs at most once. These two classes of models lead to a nontrivial family generation due to supersymmetry. For N = 8 extended supersymmetry, they give at most three and five families, respectively. The first class of models is more natural in the way it excludes SU(5) exotics. The same analysis is extended to the massless multiplets that can be obtained from bilinear composite fields of the (preonic) elementary fields of N extended supergravity. We prove that the generation of families requires the repetition of massless multiplets and that ($\text{10 +}\text{5}$) SU(5) families can only be generated in pairs. General properties of multilinear composite operators of the preonic fields are given and the rôle of massive representations to classify towers of operators with definite spin is pointed out.     

Another connection between the Λ parameters of the euclidean lattice and continuum QCD

We compute the relation between the lattice and the short-distance q$\text{q}$-force Λ parameters. The result of this perturbative computation is in striking agreement with the Creutz Monte Carlo estimate.     

Interacting boson-fermion model of collective states I,the Spin (6) limit

We discuss in detail one of the three possible spinor symmetries of the interacting boson-fermion model. This symmetry, Spin (6), arises when the bosons have SO(6) symmetry and the fermions occupy a single particle orbital with $\text{j =}\text{1}\text{2}$. We derive, within the framework of the Spin (6) symmetry, closed expressions for energies, electromagnetic (EO, M1, E2) transition rates, static moments and (one and two) nucleon transfer reaction intensities.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

Boson-Fermion duality in four dimensions: Neutrinos from photons and vice versa

We present a construction of bosons from fermions in any space dimension. In three dimensions, photons can be constructed from neutrinos, and posses the correct P, C, T and rotational properties. A new massless particle is required, identical to the photon except that it is even under charge conjugations. In a special sense, a photon is $\text{7}\text{8}$ of a neutrino. The inverse construction, neutrinos from these massless bosons, suggests that none of these is truly fundamental.     

Simulating the all-order hopping expansion II: Wilson fermions

We investigate the extension of the Prokof'ev–Svistunov worm algorithm to Wilson lattice fermions in an external scalar field. We effectively simulate by Monte Carlo the graphs contributing to the hopping expansion of the two-point function on a finite lattice to arbitrary order. Tests are conducted for a constant background field i.e. free fermions at some mass. For the method introduced here this is expected to be a representative case. Its advantage is that we know the exact answers and can thus make stringent tests on the numerics. The approach is formulated in both two and three space–time dimensions. In D=2$D=2$ Wilson fermions enjoy special positivity properties and the simulation is similarly efficient as in the Ising model. In D=3$D=3$ the method also works at sufficiently large mass, but there is a hard sign problem in the present formulation hindering us to take the continuum limit.     

Bond percolation threshold in the simple cubic lattice

In the simple cubic lattice, for bond-percolation our Monte Carlo simulation gives a value of p_∞ = 0.2492 ± 0.0002, in contradiction to a recent speculation of Sahimi et al. (p_∞ = 0.25273) and to the possibility $\text{p}{}_{\mathrm{\infty }}\text{=}\text{1}\text{4}$.