Finite Volume Effects Research Articles

Finite volume effects in thermal field theory

We analyze the effect of a finite volume on the thermodynamic potentials of a relativistic quantum field theory defined on a hypertorus at vanishing chemical potential. Using the symmetries of the Euclidean partition function, we interpret the thermodynamic observables as the expectation value of the energy-momentum tensor in the same theory living on a β × L2 volume. In the case where the screening correlation lengths in the thermal system are finite, we obtain a closed formula for the leading finite volume effects in terms of the smallest screening mass. This formula can be used to estimate, and possibly correct for, the leading finite volume effects in lattice calculations of QCD thermodynamics.

Nucleon form factors with2+1flavor dynamical domain-wall fermions

We report our numerical lattice QCD calculations of the isovector nucleon form factors for the vector and axial-vector currents: the vector, induced tensor, axial-vector, and induced pseudoscalar form factors. The calculation is carried out with the gauge configurations generated with ${N}_{f}=2+1$ dynamical domain-wall fermions and Iwasaki gauge actions at $\ensuremath{\beta}=2.13$, corresponding to a cutoff ${a}^{\ensuremath{-}1}=1.73\text{ }\text{ }\mathrm{GeV}$, and a spatial volume of $(2.7\text{ }\text{ }\mathrm{fm}{)}^{3}$. The up and down-quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange quark mass is about 12% heavier than the physical one. We calculate the form factors in the range of momentum transfers, $0.2<{q}^{2}<0.75\text{ }\text{ }{\mathrm{GeV}}^{2}$. The vector and induced tensor form factors are well described by the conventional dipole forms and result in significant underestimation of the Dirac and Pauli mean-squared radii and the anomalous magnetic moment compared to the respective experimental values. We show that the axial-vector form factor is significantly affected by the finite spatial volume of the lattice. In particular in the axial charge, ${g}_{A}/{g}_{V}$, the finite-volume effect scales with a single dimensionless quantity, ${m}_{\ensuremath{\pi}}L$, the product of the calculated pion mass and the spatial lattice extent. Our results indicate that for this quantity, ${m}_{\ensuremath{\pi}}L>6$ is required to ensure that finite-volume effects are below 1%.

Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.

Search for the charmonia dissociation in lattice QCD

Dissociation of charmonia in hot QCD is studied on a quenched anisotropic lattice using standard plaquette gauge action and O(a) improved Wilson fermion action. Simulations are carried out at temperatures in the range 0.88 TC to 2.3 TC. We calculate effective masses and Bethe–Salpeter wavefunctions for the ground state (1S, 1P) and some excited states (2S, 2P) with variational analysis. To distinguish between bound states and scattering states, we apply two methods: first, we compare effective masses for charmonium correlation functions under different spatial boundary conditions. Because the scattering states are sensitive to the boundary conditions, finite volume effects are expected when a charmonium dissociates. Second, we study if the wavefunctions show a sign of scattering states which will be characterized by a broad wavefunction. We find no clear evidence of dissociation for the charmonium states up to 2.3 TC so far.

Subhaloes in scale-free cosmologies

We explore the dependence of the subhalo mass function on the spectral index n of the linear matter power spectrum using scale-free Einstein-de Sitter simulations with n=-1 and n=-2.5. We carefully consider finite volume effects that may call into question previous simulations of n =-0.75. We infer that although the subhalo mass function appears to be independent of n so long as n>=-2, it begins to flatten as n->-3. Thus, the common practice of using alpha=-1.0 may greatly overestimate the number of subhaloes at the smallest scales in the CDM hierarchy.

Erratum: KaonB-parameter in mixed action chiral perturbation theory [Phys. Rev. D75, 034502 (2007)

We calculate the kaon B-parameter, B_K, in chiral perturbation theory for a partially quenched, mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We find that the resulting expression is similar to that in the continuum, and in fact has only two additional unknown parameters. At one-loop order, taste-symmetry violations in the staggered sea sector only contribute to flavor-disconnected diagrams by generating an O(a^2) shift to the masses of taste-singlet sea-sea mesons. Lattice discretization errors also give rise to an analytic term which shifts the tree-level value of B_K by an amount of O(a^2). This term, however, is not strictly due to taste-breaking, and is therefore also present in the expression for B_K for pure G-W lattice fermions. We also present a numerical study of the mixed B_K expression in order to demonstrate that both discretization errors and finite volume effects are small and under control on the MILC improved staggered lattices.

Volume effects for pion two-point functions in constant electric and magnetic fields

We compute finite volume effects relevant for lattice QCD simulations using quantized background fields. Focusing on constant electric and magnetic fields on a periodic lattice, we determine volume corrections to pion two-point functions using chiral perturbation theory. Among such corrections are the finite volume shifts to the electric and magnetic polarizabilities, which are numerically shown to be non-negligible. We additionally find that all terms in the single-particle effective action can be renormalized by infrared effects. This includes Born couplings to the pion current and total charge-squared, which can be renormalized due to the nature of gauge invariance on a torus.

Finite-volume effects and dynamical chiral symmetry breaking inQED3

We investigate the impact of finite-volume effects on the critical number of flavors, ${N}_{f}^{c}$, for chiral symmetry restoration in ${\text{QED}}_{3}$. To this end we solve a set of coupled Dyson-Schwinger equations on a torus. For order parameters such as the anomalous dimension of the fermion wave function or the chiral condensate, we find substantial evidence for a large dependence on the volume. We observe a shift in ${N}_{f}^{c}$ from values in the range of $3.61\ensuremath{\le}{N}_{f}^{c}\ensuremath{\le}3.84$ in the infinite-volume and continuum limit down to values below ${N}_{f}\ensuremath{\le}1.5$ at finite volumes in agreement with earlier results of Gusynin and Reenders in a simpler truncation scheme. These findings explain discrepancies in ${N}_{f}^{c}$ between continuum and lattice studies.

More on Gribov copies and propagators in Landau-gauge Yang-Mills theory

Fixing a gauge in the nonperturbative domain of Yang-Mills theory is a nontrivial problem due to the presence of Gribov copies. In particular, there are different gauges in the nonperturbative regime which all correspond to the same definition of a gauge in the perturbative domain. Gauge-dependent correlation functions may differ in these gauges. Two such gauges are the minimal Landau gauge and the absolute Landau gauge, both corresponding to the perturbative Landau gauge. These, and their numerical implementation, are described and presented in detail. Other choices will also be discussed. This investigation is performed, using numerical lattice gauge theory calculations, by comparing the propagators of gluons and ghosts for the minimal Landau gauge and the absolute Landau gauge in SU(2) Yang-Mills theory. It is found that the propagators are different in the far infrared and even at energy scales of the order of half a GeV. In particular, the finite-volume effects are also modified. This is observed in two and three dimensions. Some remarks on the four-dimensional case are provided as well.

Nucleon form factors from quenched lattice QCD with domain wall fermions

We present a quenched lattice calculation of the weak nucleon form factors: vector [${F}_{V}({q}^{2})$], induced tensor [${F}_{T}({q}^{2})$], axial vector [${F}_{A}({q}^{2})$] and induced pseudoscalar [${F}_{P}({q}^{2})$] form factors. Our simulations are performed on three different lattice sizes ${L}^{3}\ifmmode\times\else\texttimes\fi{}T={24}^{3}\ifmmode\times\else\texttimes\fi{}32$, ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$, and ${12}^{3}\ifmmode\times\else\texttimes\fi{}32$ with a lattice cutoff of ${a}^{\ensuremath{-}1}\ensuremath{\approx}1.3\text{ }\text{ }\mathrm{GeV}$ and light quark masses down to about $1/4$ the strange quark mass (${m}_{\ensuremath{\pi}}\ensuremath{\approx}390\text{ }\text{ }\mathrm{MeV}$) using a combination of the DBW2 gauge action and domain wall fermions. The physical volume of our largest lattice is about $(3.6\text{ }\text{ }\mathrm{fm}{)}^{3}$, where the finite volume effects on form factors become negligible and the lower momentum transfers (${q}^{2}\ensuremath{\approx}0.1\text{ }\text{ }{\mathrm{GeV}}^{2}$) are accessible. The ${q}^{2}$ dependences of form factors in the low ${q}^{2}$ region are examined. It is found that the vector, induced tensor, and axial-vector form factors are well described by the dipole form, while the induced pseudoscalar form factor is consistent with pion-pole dominance. We obtain the ratio of axial to vector coupling ${g}_{A}/{g}_{V}={F}_{A}(0)/{F}_{V}(0)=1.219(38)$ and the pseudoscalar coupling ${g}_{P}={m}_{\ensuremath{\mu}}{F}_{P}(0.88{m}_{\ensuremath{\mu}}^{2})=8.15(54)$, where the errors are statistical errors only. These values agree with experimental values from neutron $\ensuremath{\beta}$ decay and muon capture on the proton. However, the root mean-squared radii of the vector, induced tensor, and axial vector underestimate the known experimental values by about 20%. We also calculate the pseudoscalar nucleon matrix element in order to verify the axial Ward-Takahashi identity in terms of the nucleon matrix elements, which may be called as the generalized Goldberger-Treiman relation.

Effective Coulomb potential in SU(3) lattice Yang-Mills theory

We study the infrared behavior of the effective Coulomb potential in lattice SU(3) Yang-Mills theory in the Coulomb gauge. We use lattices up to a size of 48{sup 4} and three values of the inverse coupling, {beta}=5.8, 6.0, and 6.2. While finite-volume effects are hardly visible in the effective Coulomb potential, scaling violations and a strong dependence on the choice of Gribov copy are observed. We obtain bounds for the Coulomb string tension that are in agreement with Zwanziger's inequality relating the Coulomb string tension to the Wilson string tension.

Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation

A detailed comparison between data from experimental measurements and numerical simulations of Lagrangian velocity structure functions in turbulence is presented. Experimental data, at Reynolds number ranging from Rλ=350 to Rλ=815, are obtained in a swirling water flow between counter-rotating baffled disks. Direct numerical simulations (DNS) data, up to Rλ=284, are obtained from a statistically homogeneous and isotropic turbulent flow. By integrating information from experiments and numerics, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers is achieved. To this purpose, we discuss in detail the importance of statistical errors, anisotropy effects, and finite volume and filter effects, finite trajectory lengths. The local scaling properties of the Lagrangian velocity increments in the two data sets are in good quantitative agreement for all time lags, showing a degree of intermittency that changes if measured close to the Kolmogorov time scales or at larger time lags. This systematic study resolves apparent disagreement between observed experimental and numerical scaling properties.

Analytical Solution for the Stress Field around a Hard Spherical Particle in a Metal Matrix Composite Incorporating Size and Finite Volume Effects

An analytical solution is presented for the stress field around an elastic spherical particle embedded in a finite elastic—plastic metal matrix subjected to hydrostatic tension. The formulation is based on a strain gradient plasticity theory and a unit cell of finite dimensions, and the resulting solution incorporates the size and finite volume effects. The counterpart solutions based on classical elasticity and plasticity are reduced from the current solution, and the solution for that of an elastic spherical particle embedded in an infinitely large elastic—plastic matrix is obtained as a special case. The newly derived solution can capture the particle size effect and account for composites with both dilute and non-dilute particle distributions, unlike existing analytical models. The stress concentration factor on the particle/matrix interface, which depends on the particle size and volume fraction, is determined by directly applying the current solution. Numerical results quantitatively show that the stress concentration factor increases as the particle volume fraction increases and the strain-hardening level of the matrix material decreases. Also, these results reveal that the stress concentration factor decreases with decreasing particle size at the micron scale, thereby predicting the particle size effect.

Nucleon Axial Charge in (2+1)-Flavor Dynamical-Lattice QCD with Domain-Wall Fermions

We present results for the nucleon axial charge g{A} at a fixed lattice spacing of 1/a=1.73(3) GeV using 2+1 flavors of domain wall fermions on size 16;{3} x 32 and 24;{3} x 64 lattices (L=1.8 and 2.7 fm) with length 16 in the fifth dimension. The length of the Monte Carlo trajectory at the lightest m_{pi} is 7360 units, including 900 for thermalization. We find finite volume effects are larger than the pion mass dependence at m{pi}=330 MeV. We also find a scaling with the single variable m{pi}L which can also be seen in previous two-flavor domain wall and Wilson fermion calculations. Using this scaling to eliminate the finite-volume effect, we obtain g{A}=1.20(6)(4) at the physical pion mass, m_{pi}=135 MeV, where the first and second errors are statistical and systematic. The observed finite-volume scaling also appears in similar quenched simulations, but disappear when V>or=(2.4 fm);{3}. We argue this is a dynamical quark effect.

QED in Finite Volume and Finite Size Scaling Effect on Electromagnetic Properties of Hadrons

On account of the computation of the virtual photon corrections to the hadronic properties by means of lattice QCD simulation, we study the finite size scaling effect on the QED correction using low energy effective theory of QCD with QED. For this purpose, we begin with formulating a new QED on the space with finite volume. By adapting this formalism to the partially quenched QCD with electromagnetism, we compute the finite size scaling effect on the electromagnetic correction to the masses of pseudo-Nambu-Goldstone bosons and observe qualitative features.

Erratum: Improved Landau gauge fixing and the suppression of finite-volume effects of the lattice gluon propagator [Phys. Rev. D77, 014504 (2008)

For the gluon propagator of pure SU(2) lattice gauge theory in the Landau gauge we investigate the effect of Gribov copies and finite-volume effects. Concerning gauge fixing, we enlarge the accessible gauge orbits by adding non-periodic Z(2) gauge transformations and systematically employ the simulated annealing algorithm. Strategies to keep all Z(2) sectors under control within reasonable CPU time are discussed. We demonstrate that the finite-volume effects in the infrared regime become ameliorated. Reaching a physical volume of about (6.5 fm)^4, we find that the propagator, calculated with the indicated improvements, becomes flat in the region of smallest momenta. First signs in 4d of a final decrease towards vanishing momentum are discussed.

Observational biases in Lagrangian reconstructions of cosmic velocity fields

Lagrangian reconstruction of large-scale peculiar velocity fields can be strongly affected by observational biases. We develop a thorough analysis of these systematic effects by relying on specially selected mock catalogues. For the purpose of this paper, we use the MAK reconstruction method, although any other Lagrangian reconstruction method should be sensitive to the same problems. We extensively study the uncertainty in the mass-to-light assignment due to luminosity incompleteness, and the poorly-determined relation between mass and luminosity. The impact of redshift distortion corrections is analyzed in the context of MAK and we check the importance of edge and finite-volume effects on the reconstructed velocities. Using three mock catalogues with different average densities, we also study the effect of cosmic variance. In particular, one of them presents the same global features as found in observational catalogues that extend to 80 Mpc/h scales. We give recipes, checked using the aforementioned mock catalogues, to handle these particular observational effects, after having introduced them into the mock catalogues so as to quantitatively mimic the most densely sampled currently available galaxy catalogue of the nearby universe. Once biases have been taken care of, the typical resulting error in reconstructed velocities is typically about a quarter of the overall velocity dispersion, and without significant bias. We finally model our reconstruction errors to propose an improved Bayesian approach to measure Omega_m in an unbiased way by comparing the reconstructed velocities to the measured ones in distance space, even though they may be plagued by large errors. We show that, in the context of observational data, a nearly unbiased estimator of Omega_m may be built using MAK reconstruction.

Improved Landau gauge fixing and the suppression of finite-volume effects of the lattice gluon propagator

For the gluon propagator of pure SU(2) lattice gauge theory in the Landau gauge we investigate the effect of Gribov copies and finite-volume effects. Concerning gauge fixing, we enlarge the accessible gauge orbits by adding nonperiodic Z(2) gauge transformations and systematically employ the simulated annealing algorithm. Strategies to keep all Z(2) sectors under control within reasonable CPU time are discussed. We demonstrate that the finite-volume effects in the infrared regime become ameliorated. Reaching a physical volume of about (6.5 fm){sup 4}, we find that the propagator, calculated with the indicated improvements, becomes flat in the region of smallest momenta. There are first signs in four dimensions of a decrease towards vanishing momentum.

INFRARED-SUPPRESSED GLUON PROPAGATOR IN 4D YANG–MILLS THEORY IN A LANDAU-LIKE GAUGE

The infrared behavior of the gluon propagator is directly related to confinement in QCD. Indeed, the Gribov–Zwanziger scenario of confinement predicts an infrared vanishing (transverse) gluon propagator in Landau-like gauges, implying violation of reflection positivity and gluon confinement. Finite-volume effects make it very difficult to observe (in the minimal Landau gauge) an infrared suppressed gluon propagator in lattice simulations of the four-dimensional case. Here we report results for the SU(2) gluon propagator in a gauge that interpolates between the minimal Landau gauge (for gauge parameter λ equal to 1) and the minimal Coulomb gauge (corresponding to λ = 0). For small values of λ we find that the spatially-transverse gluon propagator D tr (0, |p|), considered as a function of the spatial momenta |p|, is clearly infrared suppressed. This result is in agreement with the Gribov–Zwanziger scenario and with previous numerical results in the minimal Coulomb gauge. We also discuss the nature of the limit λ→0 (complete Coulomb gauge) and its relation to the standard Coulomb gauge (λ = 0). Our findings are corroborated by similar results in the three-dimensional case, where the infrared suppression is observed for all considered values of λ.

The phase structure of a chirally invariant lattice Higgs-Yukawa model for small and for large values of the Yukawa coupling constant

We consider a chirally invariant lattice Higgs-Yukawa model based on the Neuberger overlap operator. As a first step towards the eventual determination of Higgs mass bounds we study the phase diagram of the model analytically in the large Nf-limit. We present an expression for the effective potential at tree-level in the regime of small Yukawa and quartic coupling constants and determine the order of the phase transitions. In the case of strong Yukawa couplings the model effectively becomes an O(4)-symmetric non-linear sigma-model for all values of the quartic coupling constant. This leads to the existence of a symmetric phase also in the regime of large values of the Yukawa coupling constant. On finite and small lattices, however, strong finite volume effects prevent the expectation value of the Higgs field from vanishing thus obscuring the existence of the symmetric phase at strong Yukawa couplings.