Fixed Scale Approach Research Articles

A basic algorithm for generating an individualized numerical scale

Linguistic labels are an effective means of expressing qualitative assessments because they account for the uncertain nature of human preferences. However, to perform computations with linguistic labels, they must first be converted into numbers using a scale function. Within the context of the Analytic Hierarchy Process (AHP), the most popular scale used to represent linguistic labels numerically is the linear 1–9 scale, which was proposed by Saaty. However, this scale has been criticized by several researchers, and various alternatives have been proposed. There is a growing interest in scale individualization rather than relying on a generic fixed scale because the perceptions of the decision maker regarding these linguistic labels are highly subjective. The methods proposed in the literature for scale individualization focus on minimizing transitivity errors (i.e., consistency). In this paper, we propose a novel, easy-to-learn, easy-to-implement, and computationally less demanding scale individualization approach based on compatibility. We also developed an experimental setup and introduced two new metrics that can be used by researchers that contribute to the theory of AHP. To assess the value of scale individualization in general and the performance of the proposed novel approach, numerical and two empirical studies were conducted. The results of the analyses demonstrate that scale individualization outperforms the conventional fixed-scale approach and validates the benefits of the proposed novel heuristic.

Hadrons at high temperature: An update from the FASTSUM collaboration

We present the most recent results from the FASTSUM collaboration for hadron properties at high temperature. This includes the temperature dependence of the light and charmed meson and baryon spectrum, as well as properties of heavy quarkonia. The results are obtained using anisotropic lattices with a fixed scale approach. We also present the status of our next generation gauge ensembles.

QCD transition at the physical point, and its scaling window from twisted mass Wilson fermions

We study the scaling properties of the finite temperature QCD phase transition, for light quark masses ranging from the heavy quark regime to their physical values. The lattice results are obtained in the fixed scale approach from simulations of Nf=2+1+1 flavours of Wilson fermions at maximal twist. We identify an order parameter free from the linear contributions in mass due to additive renormalization and regular terms in the Equation of State, which proves useful for the assessment of the hypothesized universal behaviour. We find compatibility with the 3D O(4) universality class for the physical pion mass and temperatures 120 MeV ≲T≲300 MeV. We discuss violation of scaling at larger masses and a possible cross-over to mean field behaviour. The chiral extrapolation T0=134−4+6 MeV of the pseudocritical temperature is robust against predictions of different universality classes and consistent with its estimate from the O(4) Equation of State for the physical pion mass.

Fluctuations and correlations of net baryon number, electric charge and strangeness in a background magnetic field

We present results on the second-order fluctuations of and correlations among net baryon number, electric charge and strangeness in (2+1)-flavor lattice QCD in the presence of a background magnetic field. Simulations are performed using the tree-level improved gauge action and the highly improved staggered quark (HISQ) action with a fixed scale approach ($a\simeq$ 0.117 fm). The light quark mass is set to be 1/10 of the physical strange quark mass and the corresponding pion mass is about 220 MeV at vanishing magnetic field. Simulations are performed on $32^3\times N_\tau$ lattices with 9 values of $N_\tau$ varying from 96 to 6 corresponding to temperatures ranging from zero up to 281 MeV. The magnetic field strength $eB$ is simulated with 15 different values up to $\sim$2.5 GeV$^2$ at each nonzero temperature. We find that quadratic fluctuations and correlations do not show any singular behavior at zero temperature in the current window of $eB$ while they develop peaked structures at nonzero temperatures as $eB$ grows. By comparing the electric charge-related fluctuations and correlations with hadron resonance gas model calculations and ideal gas limits we find that the changes in degrees of freedom start at lower temperatures in stronger magnetic fields. Significant effects induced by magnetic fields on the isospin symmetry and ratios of net baryon number and baryon-strangeness correlation to strangeness fluctuation are observed, which could be useful for probing the existence of a magnetic field in heavy-ion collision experiments.

Finite-temperature gluon spectral functions from N_f=2+1+1 lattice QCD

We investigate gluon correlation functions and spectral functions at finite temperature in Landau gauge on lattice QCD ensembles with N_f=2+1+1 dynamical twisted-mass quarks flavors, generated by the tmfT collaboration. They cover a temperature range from 0.8le T/T_Cle 4 using the fixed-scale approach. Our study of spectral properties is based on a novel Bayesian approach for the extraction of non-positive-definite spectral functions. For each binned spatial momentum we take into account the gluon correlation functions at all available discrete imaginary frequencies. Clear indications for the existence of a well defined quasi-particle peak are obtained. Due to a relatively small number of imaginary frequencies available, we focus on the momentum and temperature dependence of the position of this spectral feature. The corresponding dispersion relation reveals different in-medium masses for longitudinal and transversal gluons at high temperatures, qualitatively consistent with weak coupling expectations.

QCD thermodynamics with continuum extrapolated Wilson fermions. II.

We continue our investigation of 2+1 flavor QCD thermodynamics using dynamical Wilson fermions in the fixed scale approach. Two additional pion masses, approximately 440 MeV and 285 MeV, are added to our previous work at 545 MeV. The simulations were performed at 3 or 4 lattice spacings at each pion mass. The renormalized chiral condensate, strange quark number susceptibility and Polyakov loop is obtained as a function of the temperature and we observe a decrease in the light chiral pseudo-critical temperature as the pion mass is lowered while the pseudo-critical temperature associated with the strange quark number susceptibility or the Polyakov loop is only mildly sensitive to the pion mass. These findings are in agreement with previous continuum results obtained in the staggered formulation.

Fixed-scale approach to finite-temperature lattice QCD with shifted boundaries

We study the thermodynamics of the SU(3) gauge theory using the fixed-scale approach with shifted boundary conditions. The fixed-scale approach can reduce the numerical cost of the zero-temperature part in the equation of state calculations, while the number of possible temperatures is limited by the integer $N_t$, which represents the temporal lattice extent. The shifted boundary conditions can overcome such a limitation while retaining the advantages of the fixed-scale approach. Therefore, our approach enables the investigation of not only the equation of state in detail, but also the calculation of the critical temperature with increased precision even with the fixed-scale approach. We also confirm numerically that the boundary conditions suppress the lattice artifact of the equation of state, which has been confirmed in the non-interacting limit.

Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD

Two-color lattice QCD with N_f=4 staggered fermion degrees of freedom (no rooting trick is applied) with equal electric charge q is studied in a homogeneous magnetic background field B and at non-zero temperature T. In order to circumvent renormalization as a function of the bare coupling we apply a fixed-scale approach. We study the influence of the magnetic field on the critical temperature. At rather small pseudo-scalar meson mass ($m_{\pi} \approx 175 \mathrm{MeV} \approx T_c(B=0)$) we confirm a monotonic rise of the quark condensate $<\bar{\psi} \psi>$ with increasing magnetic field strength, i.e. magnetic catalysis, as long as one is staying within the confinement or deconfinement phase. In the transition region we find indications for a non-monotonic behavior of $T_c(B)$ at low magnetic field strength ($qB<0.8 \mathrm{GeV}^2$) and a clear rise at stronger magnetic field. The conjectured existence of a minimum value $T_c(B^{*}) < T_c(B=0)$ would leave a temperature window for a decrease of $<\bar{\psi} \psi>$ with rising $B$ (inverse magnetic catalysis) also in the present model.

Fractality and other properties of center domains at finite temperature: SU(3) lattice gauge theory

Using finite temperature SU(3) lattice gauge theory in the fixed scale approach we analyze center properties of the local Polyakov loop $L(x)$. We construct spatial clusters of points $x$ where the phase of $L(x)$ is near the same center element and study their properties as a function of temperature. We find that below the deconfinement transition the clusters form objects with a fractal dimension $D&lt;3$. As the temperature is increased, the largest cluster starts to percolate and its dimensionality approaches $D=3$. The fractal structure of the clusters in the transition region may have implications regarding both the small shear viscosity and the large opacity of the quark gluon plasma observed in heavy-ion collision experiments.

Application of Fixed Scale Approach to Static Quark Free Energies in Quenched and 2 + 1 Flavor Lattice QCD with Improved Wilson Quark Action

Free energies between static quarks and Debye screening masses in the quark-gluon plasma are studied on the basis of Polyakov-line correlations in lattice simulations of 2+1 flavors QCD with the renormalization-group improved gluon action and the $O(a)$-improved Wilson quark action. We perform simulations at $m_{\rm PS}/m_{\rm V} = 0.63$ (0.74) for light (strange) flavors with lattice sizes of $32^3 \times N_t$ with $N_t=4$--12. We adopt the fixed-scale approach, where temperature can be varied without changing the spatial volume and renormalization factor. We find that, at short distance, the free energies of static quarks in color-singlet channel converge to the static-quark potential evaluated from the Wilson-loop at zero-temperature, in accordance with the expected insensitivity of short distance physics to the temperature. At long distance, the free energies of static quarks approach to twice the single-quark free energies, implying that the interaction between static quarks is fully screened. The screening properties can be well described by the screened Coulomb form with appropriate Casimir factor at high temperature. We also discuss a limitation of the fixed-scale approach at high temperature.

QCD thermodynamics with continuum extrapolated Wilson fermions I

QCD thermodynamics is considered using Wilson fermions in the fixed scale approach. The temperature dependence of the renormalized chiral condensate, quark number susceptibility and Polyakov loop is measured at four lattice spacings allowing for a controlled continuum limit. The light quark masses are fixed to heavier than physical values in this first study. Finite volume effects are ensured to be negligible by using approriately large box sizes. The final continuum results are compared with staggered fermion simulations performed in the fixed N_t approach. The same continuum renormalization conditions are used in both approaches and the final results agree perfectly.

Equation of state in2+1flavor QCD with improved Wilson quarks by the fixed scale approach

We study the equation of state in 2+1 flavor QCD with nonperturbatively improved Wilson quarks coupled with the RG-improved Iwasaki glue. We apply the $T$-integration method to nonperturbatively calculate the equation of state by the fixed-scale approach. With the fixed-scale approach, we can purely vary the temperature on a line of constant physics without changing the system size and renormalization constants. Unlike the conventional fixed-$N_t$ approach, it is easy to keep scaling violations small at low temperature in the fixed scale approach. We study 2+1 flavor QCD at light quark mass corresponding to $m_\pi/m_\rho \simeq 0.63$, while the strange quark mass is chosen around the physical point. Although the light quark masses are heavier than the physical values yet, our equation of state is roughly consistent with recent results with highly improved staggered quarks at large $N_t$.

Ab initio study of the thermodynamics of quantum chromodynamics on the lattice at zero and finite densities

The WHOT-QCD Collaboration is pushing forward a series of lattice studies of quantum chromodynamics (QCD) at finite temperatures and densities using improvedWilson quarks. Because Wilson-type quarks require more computational resources than the more widely used staggeredtype quarks, various theoretical and computational techniques have to be developed and applied. In this paper, we introduce a fixed-scale approach armed with a T -integration method, a Gaussian method based on cumulant expansion, and a histogram method combined with the reweighting technique. Using these methods, we have carried out the first study of finite-density QCD with Wilson-type quarks and the first calculation of the equation of state with 2 + 1 flavors of Wilson-type quarks.We present the results of these studies and discuss perspectives towards clarification of the properties of 2 + 1 flavor QCD at the physical point at finite temperatures and densities.

QCD Thermodynamics at Zero and Finite Densities with Improved Wilson Quarks

The WHOT-QCD Collaboration is pushing forward lattice studies of QCD at finite temperatures and densities using improved Wilson quarks. We first present results on QCD at zero and finite densities with two flavors of degenerate quarks (N_F=2 QCD) adopting the conventional fixed-Nt approach. We then report on the status of a study of N_F=2+1 QCD adopting a fixed-scale approach armed with the T-integration method which we have developed.

Renormalized Polyakov loop in the fixed scale approach

I compute the deconfinement order parameter for the SU(2) lattice gauge theory, the Polyakov loop, using the fixed scale approach for two different scales and show how one can obtain a physical, renormalized, order parameter. The generalization to other gauge theories, including quenched or full QCD, is straightforward.

Free energies of heavy quarks in full-QCD lattice simulations with Wilson-type quark action

The free energy between a static quark and an antiquark is studied by using the color-singlet Polyakov-line correlation at finite temperature in lattice QCD with 2+1 flavors of improved Wilson quarks. From the simulations on 32 3 × 12, 10, 8, 6, 4 lattices in the high temperature phase, based on the fixed scale approach, we find that, the heavy-quark free energies at short distance converge to the heavy-quark potential evaluated from the Wilson loop at zero temperature, in accordance with the expected insensitivity of short distance physics to the temperature. At long distance, the heavy-quark free energies approach to twice the single-quark free energies, implying that the interaction between heavy quarks is screened. The Debye screening mass obtained from the long range behavior of the free energy is compared with the results of thermal perturbation theory.

Fixed scale approach to the equation of state on the lattice

We propose a fixed scale approach to calculate the equation of state (EOS) in lattice QCD. In this approach, the temperature T is varied by N t at fixed lattice spacings. This enables us to reduce T = 0 simulations which are required to provide basic data in finite temperature studies but are quite expensive in the conventional fixed- N t approach. Since the conventional integral method to obtain the pressure is inapplicable at fixed scale, we introduce a new method, “T-integration method”, to calculate pressure non-perturbatively. We test the fixed scale approach armed with the T-integral method in quenched QCD on isotropic and anisotropic lattices. Our method is found to be powerful to obtain reliable results for the equation of state, especially at intermediate and low temperatures. Reduction of the computational cost of T = 0 simulations is indispensable to study EOS in QCD with dynamical quarks. The status of our study in N f = 2 + 1 QCD with improved Wilson quarks is also reported.

Fixed scale approach to equation of state in lattice QCD

A new approach to study the equation of state in finite-temperature QCD is proposed on the lattice. Unlike the conventional method in which the temporal lattice size ${N}_{t}$ is fixed, the temperature $T$ is varied by changing ${N}_{t}$ at the fixed lattice scale. The pressure of the hot QCD plasma is calculated by the integration of the trace anomaly with respect to $T$ at the fixed lattice scale. This ``$T$-integral method'' is tested in quenched QCD on isotropic and anisotropic lattices and is shown to give reliable results, especially at intermediate and low temperatures.

High energy hadron spin observables

The free energy between a static quark and an antiquark is studied by using the color-singlet Polyakov-line correlation at finite temperature in lattice QCD with 2+1 flavors of improved Wilson quarks. From the simulations on 323 × 12, 10, 8, 6, 4 lattices in the high temperature phase, based on the fixed scale approach, we find that, the heavy-quark free energies at short distance converge to the heavy-quark potential evaluated from the Wilson loop at zero temperature, in accordance with the expected insensitivity of short distance physics to the temperature. At long distance, the heavy-quark free energies approach to twice the single-quark free energies, implying that the interaction between heavy quarks is screened. The Debye screening mass obtained from the long range behavior of the free energy is compared with the results of thermal perturbation theory.

Fixed-scale transformation approach to linear and branched polymers

The radius exponent of two- and three-dimensional self-avoiding walks and branched polymers are computed in the fixed-scale transformation framework. The method requires the knowledge of the critical fugacity kc, but from this non-universal parameter it is possible to compute the universal critical exponent. The results obtained are within 1% of exact or numerical values. This confirms the versatility and quantitative power of this new theoretical approach and gives the opportunity to provide a discussion of the analogies and differences between the real space renormalization group and the fixed-scale transformation method.