Lattice Artifacts Research Articles

Continuum extrapolation of Ward identities in {{mathcal {N}}=1} supersymmetric SU(3) Yang\u2013Mills theory

In {mathcal {N}}=1 supersymmetric Yang–Mills theory, regularised on a space-time lattice, in addition to the breaking by the gluino mass term, supersymmetry is broken explicitly by the lattice regulator. In addition to the parameter tuning in the theory, the supersymmetric Ward identities can be used as a tool to investigate lattice artefacts as well as to check whether supersymmetry can be recovered in the chiral and continuum limits. In this paper we present the numerical results of an analysis of the supersymmetric Ward identities for our available gauge ensembles at different values of the inverse gauge coupling beta and of the hopping parameter kappa . The results clearly indicate that the lattice artefacts vanish in the continuum limit, confirming the restoration of supersymmetry.

Scale setting for N_mathrm{f}=3+1 QCD

We present the scale setting for a new set of gauge configurations generated with N_mathrm{f}=3+1 Wilson quarks with a non-perturbatively determined clover coefficient in a massive O(a) improvement scheme. The three light quarks are degenerate, with the sum of their masses being equal to its value in nature and the charm quark has its physical mass. We use open boundary conditions in time direction to avoid the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. The decoupling of charm at low energy allows us to set the scale by measuring the value of the low-energy quantity t_0^star /a^2, which is the flow scale t_0 at our mass point, and comparing it to an N_mathrm{f}=2+1 result in physical units. We present the details of the algorithmic setup and tuning procedure and give the bare parameters of ensembles with two lattice spacings a=0.054fm and a=0.043fm. We discuss finite volume effects and lattice artifacts and present physical results for the charmonium spectrum. In particular the hyperfine splitting between the eta _c and J/psi mesons agrees very well with its physical value.

Topological vacuum structure of the Schwinger model with matrix product states

We numerically study the single-flavor Schwinger model with a topological $\theta$-term, which is practically inaccessible by standard lattice Monte Carlo simulations due to the sign problem. By using numerical methods based on tensor networks, especially the one-dimensional matrix product states, we explore the non-trivial $\theta$-dependence of several lattice and continuum quantities in the Hamiltonian formulation. In particular, we compute the ground-state energy, the electric field, the chiral fermion condensate, and the topological vacuum susceptibility for positive, zero, and even negative fermion mass. In the chiral limit, we demonstrate that the continuum model becomes independent of the vacuum angle $\theta$, thus respecting CP invariance, while lattice artifacts still depend on $\theta$. We also confirm that negative masses can be mapped to positive masses by shifting $\theta\rightarrow \theta +\pi$ due to the axial anomaly in the continuum, while lattice artifacts non-trivially distort this mapping. This mass regime is particularly interesting for the (3+1)-dimensional QCD analog of the Schwinger model, the sign problem of which requires the development and testing of new numerical techniques beyond the conventional Monte Carlo approach.

Machine-learning prediction for quasiparton distribution function matrix elements

There have been rapid developments in the direct calculation in lattice QCD (LQCD) of the Bjorken-$x$ dependence of hadron structure through large-momentum effective theory (LaMET). LaMET overcomes the previous limitation of LQCD to moments (that is, integrals over Bjorken-$x$) of hadron structure, allowing LQCD to directly provide the kinematic regions where the experimental values are least known. LaMET requires large-momentum hadron states to minimize its systematics and allow us to reach small-$x$ reliably. This means that very fine lattice spacing to minimize lattice artifacts at order $(P_z a)^n$ will become crucial for next-generation LaMET-like structure calculations. Furthermore, such calculations require operators with long Wilson-link displacements (in finer lattice units), increasing the communication costs relative to that of the propagator inversion. In this work, we explore whether machine-learning (ML) algorithms can make correlator predictions to reduce the computational cost of these LQCD calculations. We consider two algorithms, gradient-boosting decision tree and linear models, applied to LaMET data, the matrix elements needed to determine the kaon and $\eta_s$ unpolarized parton distribution functions (PDFs), meson distribution amplitude (DA), and the nucleon gluon PDF. We find that both algorithms can reliably predict the target observables with different fit quality and systematic errors. The predictions from smaller displacement $z$ to larger ones work better than those for momentum $p$ due to the higher correlation among the data.

Hadron\u2013Hadron interactions from N_mathrm{{f}}=2+1+1 lattice QCD: the rho ,-resonance

We present a lattice QCD investigation of the rho -meson with N_mathrm{{f}}=2+1+1 dynamical quark flavours for the first time. The calculation is performed based on gauge configuration ensembles produced by the ETM Collaboration with three lattice spacing values and pion masses ranging from 230 to 500 MeV. Applying the Lüscher method phase-shift curves are determined for all ensembles separately. Assuming a Breit–Wigner form, the rho -meson mass and width are determined by a fit to these phase-shift curves. Mass and width combined are then extrapolated to the chiral limit, while lattice artefacts are not detectable within our statistical uncertainties. For the rho -meson mass extrapolated to the physical point we find good agreement with experiment. The corresponding decay width differs by about two standard deviations from the experimental value.

Two-colour QCD phases and the topology at low temperature and high density

We delineate equilibrium phase structure and topological charge distribution of dense two-colour QCD at low temperature by using a lattice simulation with two-flavour Wilson fermions that has a chemical potential μ and a diquark source j incorporated. We systematically measure the diquark condensate, the Polyakov loop, the quark number density and the chiral condensate with improved accuracy and j → 0 extrapolation over earlier publications; the known qualitative features of the low temperature phase diagram, which is composed of the hadronic, Bose-Einstein condensed (BEC) and BCS phases, are reproduced. In addition, we newly find that around the boundary between the hadronic and BEC phases, nonzero quark number density occurs even in the hadronic phase in contrast to the prediction of the chiral perturbation theory (ChPT), while the diquark condensate approaches zero in a manner that is consistent with the ChPT prediction. At the highest μ, which is of order the inverse of the lattice spacing, all the above observables change drastically, which implies a lattice artifact. Finally, at temperature of order 0.45Tc, where Tc is the chiral transition temperature at zero chemical potential, the topological susceptibility is calculated from a gradient-flow method and found to be almost constant for all the values of μ ranging from the hadronic to BCS phase. This is a contrast to the case of 0.89Tc in which the topological susceptibility becomes small as the hadronic phase changes into the quark-gluon plasma phase.

Cuprates and center vortices: A QCD confinement mechanism in a high-T[formula omitted] context

It is suggested that the center vortex confinement mechanism, familiar in hadronic physics, may have some relevance to high-Tc phenomena. We focus specifically on the transition from the superconducting phase to the pseudogap phase. There is evidence of a vortex liquid in the latter phase, in which the pairing responsible for superconductivity still exists, but superconductivity itself does not. An analogy, drawn from particle physics, may be the Higgs to confinement phase transition in an SU(N) gauge theory, where the confined phase is a vortex liquid, and the Higgs phase is a phase of a broken global ZN symmetry. We illustrate this idea with numerical simulations of a spatially asymmetric U(1) gauge-Higgs model, with lattice artifact monopoles suppressed. We show the existence of a Higgs (superconductor) to confinement (vortex liquid) phase, explicitly identifying vortices in lattice configurations generated in the confined phase, and showing that they produce an area-law falloff in planar Wilson loops, which may be measurable experimentally. The superconducting phase is a phase of broken global Z2 symmetry.

The Leutwyler–Smilga relation on the lattice

According to the Leutwyler–Smilga relation, in Quantum Chromodynamics (QCD), the topological susceptibility vanishes linearly with the quark masses. Calculations of the topological susceptibility in the context of lattice QCD, extrapolated to zero quark masses, show a remnant nonzero value as a lattice artefact. Employing the Atiyah–Singer theorem in the framework of Symanzik’s effective action and chiral perturbation theory, we show the validity of the Leutwyler–Smilga relation in lattice QCD with lattice artefacts of order a2 in the lattice spacing a.

Absence of chiral symmetry breaking in Thirring models in 1+2 dimensions

The Thirring model is an interacting fermion theory with current-current interaction. The model in $1+2$ dimensions has applications in condensed-matter physics to describe the electronic excitations of Dirac materials. Earlier investigations with Schwinger-Dyson equations, the functional renormalization group and lattice simulations with staggered fermions suggest that a critical number of (reducible) flavors $N^{\mathrm{c}}$ exists, below which chiral symmetry can be broken spontaneously. Values for $N^{\mathrm{c}}$ found in the literature vary between $2$ and $7$. Recent lattice studies with chirally invariant SLAC fermions have indicated that chiral symmetry is unbroken for all integer flavor numbers [Wellegehausen et al., 2017]. An independent simulation based on domain wall fermions seems to favor a critical flavor-number that satisfies $1<N^{\mathrm{c}}<2$ [Hands, 2018]. However, in the latter simulations difficulties in reaching the massless limit in the broken phase (at strong coupling and after the $L_s\to\infty$ limit has been taken) are encountered. To find an accurate value $N^{\mathrm{c}}$ we study the Thirring model (by using an analytic continuation of the parity even theory to arbitrary real $N$) for $N$ between $0.5$ and $1.1$. We investigate the chiral condensate, the spectral density of the Dirac operator, the spectrum of (would-be) Goldstone bosons and the variation of the filling-factor and conclude that the critical flavor number is $N^{\mathrm{c}}=0.80(4)$. Thus we see no chiral symmetry breaking in all Thirring models with $1$ or more flavors of ($4$-component) fermions. Besides the artifact transition to the unphysical lattice artifact phase we find strong evidence for a hitherto unknown phase transition that exists for $N>N^{\mathrm{c}}$ and should answer the question of where to construct a continuum limit.

Galilean invariance restoration on the lattice

We consider the breaking of Galilean invariance due to different lattice cutoff effects in moving frames and a nonlocal smearing parameter which is used in the construction of the nuclear lattice interaction. The dispersion relation and neutron-proton scattering phase shifts are used to investigate the Galilean invariance breaking effects and ways to restore it. For $S$-wave channels, ${}^1S_0$ and ${}^3S_1$, we present the neutron-proton scattering phase shifts in moving frames calculated using both L\"uscher's formula and the spherical wall method, as well as the dispersion relation. For the $P$ and $D$ waves, we present the neutron-proton scattering phase shifts in moving frames calculated using the spherical wall method. We find that the Galilean invariance breaking effects stemming from the lattice artifacts partially cancel those caused by the nonlocal smearing parameter. Due to this cancellation, the Galilean invariance breaking effect is small, and the Galilean invariance can be restored by introducing Galilean invariance restoration operators.

Neutron-Antineutron Oscillations from Lattice QCD.

Fundamental symmetry tests of baryon number violation in low-energy experiments can probe beyond the standard model (BSM) explanations of the matter-antimatter asymmetry of the Universe. Neutron-antineutron oscillations are predicted to be a signature of many baryogenesis mechanisms involving low-scale baryon number violation. This Letter presents first-principles calculations of neutron-antineutron matrix elements needed to accurately connect measurements of the neutron-antineutron oscillation rate to constraints on |ΔB|=2 baryon number violation in BSM theories. Several important systematic uncertainties are controlled by using a state-of-the-art lattice gauge field ensemble with physical quark masses and approximate chiral symmetry, performing nonperturbative renormalization with perturbative matching to the modified minimal subtraction scheme, and studying excited state effects in two-state fits. Phenomenological implications are highlighted by comparing expected bounds from proposed neutron-antineutron oscillation experiments to predictions of a specific model of postsphaleron baryogenesis. Quantum chromodynamics is found to predict at least an order of magnitude more events in neutron-antineutron oscillation experiments than previous estimates based on the "MIT bag model" for fixed BSM parameters. Lattice artifacts and other systematic uncertainties that are not controlled in this pioneering calculation are not expected to significantly change this conclusion.

Topological susceptibility and η′ meson mass from Nf=2 lattice QCD at the physical point

In this paper we explore the computation of topological susceptibility and $\eta'$ meson mass in $N_f=2$ flavor QCD using lattice techniques with physical value of the pion mass as well as larger pion mass values. We observe that the physical point can be reached without a significant increase in the statistical noise. The mass of the $\eta'$ meson can be obtained from both fermionic two point functions and topological charge density correlation functions, giving compatible results. With the pion mass dependence of the $\eta'$ mass being flat we arrive at $M_{\eta'}= 772(18)\ \mathrm{MeV}$ without an explicit continuum limit. For the topological susceptibility we observe a linear dependence on $M_\pi^2$, however, with an additional constant stemming from lattice artifacts.

Discretization in Noncommutative Field Theory

A discretization scheme provided by the noncommutativity of space is reviewed. In the representation chosen here the radial coordinate is rendered discrete, allowing fields to be put on a lattice in a natural way. Noncommutativity is traded for a controllable type of nonlocality of the field dynamics, which in turn allows fermions to be free of lattice artefacts. Exact, singularity-free solutions are found interpreted, and their continuum limit is well-defined.

Theoretical properties of entropy in a strong coupling region

Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two-dimensional (2D) lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in 2D conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in 2D model at the large N limit.

Adiabatic continuity and confinement in supersymmetric Yang-Mills theory on the lattice

This work is a step towards merging the ideas that arise from semi-classical methods in continuum QFT with analytic/numerical lattice field theory. In this context, we consider Yang-Mills theories coupled to fermions transforming in the adjoint representation of the gauge group. These theories have the remarkable property that confinement and discrete chiral symmetry breaking can persist at weak coupling on ℝ3 × S1 up to small (non-thermal) compactification radii. This work presents a lattice investigation of a gauge theory coupled to a single adjoint Majorana fermion, the mathcal{N}=1 Supersymmetric Yang-Mills theory (SYM), and opens the prospect to understand analytically a number of non-perturbative phenomena, such as confinement, mass gap, chiral and center symmetry realizations, both on the lattice and in the continuum. We study the compactification of mathcal{N}=1 SYM on the lattice with periodic and thermal boundary conditions applied to the fermion field. We provide numerical evidences for the conjectured absence of phase transitions with periodic boundary conditions for sufficiently light lattice fermions (stability of center-symmetry), for the suppression of the chiral transition, and we provide also a diagnostic for Abelian vs. non-Abelian confinement, based on per-site Polyakov loop eigenvalue distribution functions. We identify the role of the lattice artefacts that become relevant in the very small radius regime, and we resolve some puzzles in the naive comparison between continuum and lattice.

Determination of the J/ψ chromoelectric polarizability from lattice data

The chromoelectric polarizability of $J/\ensuremath{\psi}$ is extracted from lattice QCD data on the nucleon-$J/\ensuremath{\psi}$ potential in the heavy quark limit. The value of $\ensuremath{\alpha}(1S)=(1.6\ifmmode\pm\else\textpm\fi{}0.8)\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}3}$ is obtained. This value may have a systematic uncertainty due to lattice artifacts which cannot be estimated at present, but will become controllable in future studies. We also comment on the possibility of hadrocharmonia.

Symanzik improvement with dynamical charm: a 3+1 scheme for Wilson quarks

We discuss the problem of lattice artefacts in QCD simulations enhanced by the introduction of dynamical charmed quarks. In particular, we advocate the use of a massive renormalization scheme with a close to realistic charm mass. To maintain O(a) improvement for Wilson type fermions in this case we define a finite size scheme and carry out a nonperturbative estimation of the clover coefficient csw. It is summarized in a fit formula csw(g02) that defines an improved action suitable for future dynamical charm simulations.

Analysis of Ward identities in supersymmetric Yang\u2013Mills theory

In numerical investigations of supersymmetric Yang–Mills theory on a lattice, the supersymmetric Ward identities are valuable for finding the critical value of the hopping parameter and for examining the size of supersymmetry breaking by the lattice discretisation. In this article we present an improved method for the numerical analysis of supersymmetric Ward identities, which takes into account the correlations between the various observables involved. We present the first complete analysis of supersymmetric Ward identities in mathcal {N}=1 supersymmetric Yang–Mills theory with gauge group SU(3). The results indicate that lattice artefacts scale to zero as O(a^2) towards the continuum limit in agreement with theoretical expectations.

Observation of deconfinement in a cold dense quark medium

In this paper we study the confinement/deconfinement transition in lattice SU(2) QCD at finite quark density and zero temperature. The simulations are performed on an 324 lattice with rooted staggered fermions at a lattice spacing a = 0.044 fm. This small lattice spacing allowed us to reach very large baryon density (up to quark chemical potential μq > 2000 MeV) avoiding strong lattice artifacts. In the region μq ∼ 1000 MeV we observe for the first time the confinement/deconfinement transition which manifests itself in rising of the Polyakov loop and vanishing of the string tension σ. After the deconfinement is achieved at μq > 1000 MeV, we observe a monotonous decrease of the spatial string tension σs which ends up with σs vanishing at μq > 2000 MeV. From this observation we draw the conclusion that the confinement/deconfinement transition at finite density and zero temperature is quite different from that at finite temperature and zero density. Our results indicate that in very dense matter the quark-gluon plasma is in essence a weakly interacting gas of quarks and gluons without a magnetic screening mass in the system, sharply different from a quark-gluon plasma at large temperature.

Blockspin renormalization-group study of color confinement due to violation of the non-Abelian Bianchi identity

Block-spin transformation of topological defects is applied to the violation of the non-Abelian Bianchi identity (VMABI) on lattice defined as Abelian monopoles. To get rid of lattice artifacts, we introduce various techniques smoothing the vacuum. The effective action can be determined by adopting the inverse Monte-Carlo method. The coupling constants $F(i)$ of the effective action depend on the coupling of the lattice action $\beta$ and the number of the blocking step $n$. But it is found that $F(i)$ satisfy a beautiful scaling, that is, they are a function of the product $b=na(\beta)$ alone for lattice coupling constants $3.0\le\beta\le3.9$ and the steps of blocking $1\le n\le 12$. The effective action showing the scaling behavior can be regarded as an almost perfect action corresponding to the continuum limit, since $a\to 0$ as $n\to\infty$ for fixed $b$. The almost perfect action showing the scaling is found to be independent of the smooth gauges adopted here. Then we compare the results with those obtained by the analytic blocking method of topological defects from the continuum. The infrared monopole action can be transformed into that of the string model. The physical string tension and the lowest glueball mass can be evaluated \textit{analytically} by the strong-coupling expansion of the string model. We get $\sqrt{\sigma}\simeq 1.3\sqrt{\sigma_{phys}}$ for $b\ge 1.0\ \ (\sigma_{phys}^{-1/2})$, whereas the scalar glueball mass is kept to be near $M(0^{++})\sim 3.7\sqrt{\sigma_{phys}}$. Also we can almost reproduce \textit{analytically} the scaling function of the squared monopole density determined numerically for large $b$ region $b>1.2\ (\sigma_{phys}^{-1/2})$.