Heavy quark action on the anisotropic lattice

We investigate the $O(a)$ improved quark action on anisotropic lattice as a potential framework for the heavy quark, which may enable precision computation of hadronic matrix elements of heavy-light mesons. The relativity relations of heavy-light mesons as well as of heavy quarkonium are examined on a quenched lattice with spatial lattice cutoff $a_\sigma^{-1} \simeq$ 1.6 GeV and the anisotropy $\xi=4$. We find that the bare anisotropy parameter tuned for the massless quark describes both the heavy-heavy and heavy-light mesons within 2% accuracy for the quark mass $a_\sigma m_Q < 0.8$, which covers the charm quark mass. This bare anisotropy parameter also successfully describes the heavy-light mesons in the quark mass region $a_\sigma m_Q \leq 1.2$ within the same accuracy. Beyond this region, the discretization effects seem to grow gradually. The anisotropic lattice is expected to extend by a factor $\xi$ the quark mass region in which the parameters in the action tuned for the massless limit are applicable for heavy-light systems with well controlled systematic errors.Comment: 11 pages, REVTeX4, 11 eps figure

Numerical study of O(a) improved Wilson quark action on anisotropic lattice

The $O(a)$ improved Wilson quark action on the anisotropic lattice is investigated. We carry out numerical simulations in the quenched approximation at three values of lattice spacing ($a_{\sigma}^{-1}=1$--2 GeV) with the anisotropy $\xi=a_{\sigma}/a_{\tau}=4$, where $a_{\sigma}$ and $a_{\tau}$ are the spatial and the temporal lattice spacings, respectively. The bare anisotropy $\gamma_F$ in the quark field action is numerically tuned by the dispersion relation of mesons so that the renormalized fermionic anisotropy coincides with that of gauge field. This calibration of bare anisotropy is performed to the level of 1 % statistical accuracy in the quark mass region below the charm quark mass. The systematic uncertainty in the calibration is estimated by comparing the results from different types of dispersion relations, which results in 3 % on our coarsest lattice and tends to vanish in the continuum limit. In the chiral limit, there is an additional systematic uncertainty of 1 % from the chiral extrapolation. Taking the central value $\gamma_F=\gamma_F^*$ from the result of the calibration, we compute the light hadron spectrum. Our hadron spectrum is consistent with the result by UKQCD Collaboration on the isotropic lattice. We also study the response of the hadron spectrum to the change of anisotropic parameter, $\gamma_F \to \gamma_F^* + \delta\gamma_F$. We find that the change of $\gamma_F$ by 2 % induces a change of 1 % in the spectrum for physical quark masses. Thus the systematic uncertainty on the anisotropic lattice, as well as the statistical one, is under control.Comment: 27 pages, 25 eps figures, LaTe

Heavy Quarks on Anisotropic Lattices: The Charmonium Spectrum

We present results for the mass spectrum of $c{\bar c}$ mesons simulated on anisotropic lattices where the temporal spacing $a_t$ is only half of the spatial spacing $a_s$. The lattice QCD action is the Wilson gauge action plus the clover-improved Wilson fermion action. The two clover coefficients on an anisotropic lattice are estimated using mean links in Landau gauge. The bare velocity of light $\nu_t$ has been tuned to keep the anisotropic, heavy-quark Wilson action relativistic. Local meson operators and three box sources are used in obtaining clear statistics for the lowest lying and first excited charmonium states of $^1S_0$, $^3S_1$, $^1P_1$, $^3P_0$ and $^3P_1$. The continuum limit is discussed by extrapolating from quenched simulations at four lattice spacings in the range 0.1 - 0.3 fm. Results are compared with the observed values in nature and other lattice approaches. Finite volume effects and dispersion relations are checked.Comment: 36 pages, 6 figur

Accurate Scale Determinations for the Wilson Gauge Action

Accurate determinations of the physical scale of a lattice action are required to check scaling and take the continuum limit. We present a high statistics study of the static potential for the SU(3) Wilson gauge action on coarse lattices ($5.54 \leq \beta \leq 6.0$). Using an improved analysis procedure we determine the string tension and the Sommer scale $r_0$ (and related quantities) to 1% accuracy, including all systematic errors. Combining our results with earlier ones on finer lattices, we present parameterizations of these quantities that should be accurate to about 1% for $5.6 \leq \beta \leq 6.5$. We estimate the \La-parameter of quenched QCD to be \La_\MSb = 247(16) MeV.Comment: 18 pages, LaTeX, 5 ps files (corrected typo in table 5, updated references

Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]Comment: 16 pages, LaTeX file, no figures. Revision has minor change

Improving Lattice Quark Actions

We explore the first stage of the Symanzik improvement program for lattice Dirac fermions, namely the construction of doubler-free, highly improved classical actions on isotropic as well as anisotropic lattices (where the temporal lattice spacing, a_t, is smaller than the spatial one). Using field transformations to eliminate doublers, we derive the previously presented isotropic D234 action with O(a^3) errors, as well as anisotropic D234 actions with O(a^4) or O(a_t^3, a^4) errors. Besides allowing the simulation of heavy quarks within a relativistic framework, anisotropic lattices alleviate potential problems due to unphysical branches of the quark dispersion relation (which are generic to improved actions), facilitate studies of lattice thermodynamics, and allow accurate mass determinations for particles with bad signal/noise properties, like glueballs and P-state mesons. We also show how field transformations can be used to completely eliminate unphysical branches of the dispersion relation. Finally, we briefly discuss future steps in the improvement program.Comment: Tiny changes to agree with version to appear in Nucl. Phys. B (33 pages, LaTeX, 13 eps files

Entropic C-theorems in free and interacting two-dimensional field theories

The relative entropy in two-dimensional field theory is studied on a cylinder geometry, interpreted as finite-temperature field theory. The width of the cylinder provides an infrared scale that allows us to define a dimensionless relative entropy analogous to Zamolodchikov's $c$ function. The one-dimensional quantum thermodynamic entropy gives rise to another monotonic dimensionless quantity. I illustrate these monotonicity theorems with examples ranging from free field theories to interacting models soluble with the thermodynamic Bethe ansatz. Both dimensionless entropies are explicitly shown to be monotonic in the examples that we analyze.Comment: 34 pages, 3 figures (8 EPS files), Latex2e file, continuation of hep-th/9710241; rigorous analysis of sufficient conditions for universality of the dimensionless relative entropy, more detailed discussion of the relation with Zamolodchikov's theorem, references added; to appear in Phys. Rev.

Scaling Laws and Effective Dimension in Lattice SU(2) Yang-Mills Theory with a Compactified Extra Dimension

Monte Carlo simulations are performed in a five-dimensional lattice SU(2) Yang-Mills theory with a compactified extra dimension, and scaling laws are studied. Our simulations indicate that as the compactification radius $R$ decreases, the confining phase spreads more and more to the weak coupling regime, and the effective dimension of the theory changes gradually from five to four. Our simulations also indicate that the limit $a_4 to 0$ with $R/a_4$ kept fixed exists both in the confining and deconfining phases if $R/a_4$ is small enough, where $a_4$ is the lattice spacing in the four-dimensional direction. We argue that the color degrees of freedom in QCD are confined only for $R < R_{\rm max}$, where a rough estimate shows that $1/R_{\rm max}$ lies in the TeV range. Comments on deconstructing extra dimensions are given.Comment: 15 pages, TeX, 5 figure

The (LATTICE) QCD Potential and Running Coupling: How to Accurately Interpolate between Multi-Loop QCD and the String Picture

We present a simple parameterization of a running coupling constant, defined via the static potential, that interpolates between 2-loop QCD in the UV and the string prediction in the IR. Besides the usual \Lam-parameter and the string tension, the coupling depends on one dimensionless parameter, determining how fast the crossover from UV to IR behavior occurs (in principle we know how to take into account any number of loops by adding more parameters). Using a new Ansatz for the LATTICE potential in terms of the continuum coupling, we can fit quenched and unquenched Monte Carlo results for the potential down to ONE lattice spacing, and at the same time extract the running coupling to high precision. We compare our Ansatz with 1-loop results for the lattice potential, and use the coupling from our fits to quantitatively check the accuracy of 2-loop evolution, compare with the Lepage-Mackenzie estimate of the coupling extracted from the plaquette, and determine Sommer's scale $r_0$ much more accurately than previously possible. For pure SU(3) we find that the coupling scales on the percent level for $\beta\geq 6$.Comment: 47 pages, incl. 4 figures in LaTeX [Added remarks on correlated vs. uncorrelated fits in sect. 4; corrected misprints; updated references.

Glueball Properties at Finite Temperature in SU(3) Anisotropic Lattice QCD

The thermal properties of the glueballs are studied using SU(3) anisotropic lattice QCD with beta=6.25, the renormalized anisotropy xi=a_s/a_t=4 over the lattice of the size 20^3\times N_t with N_t = 24, 26, 28, 30, 33, 34, 35, 36, 37, 38, 40, 43, 45, 50, 72 at the quenched level. To construct a suitable operator on the lattice, we adopt the smearing method, and consider its physical meaning in terms of the operator size. First, we construct the temporal correlators G(t) for the 0^{++} and 2^{++} glueballs, using more than 5,000 gauge configurations at each temperature. We then measure the pole-mass of the thermal glueballs from G(t). For the lowest 0^{++} glueball, we observe a significant pole-mass reduction of about 300 MeV near T_c or m_G(T\simeq T_c) \simeq 0.8 m_G(T\sim 0), while its size remains almost unchanged as rho(T) \simeq 0.4fm. Finally, for completeness, as an attempt to take into account the effect of thermal width Gamma(T) at finite temperature, we perform a more general new analysis of G(t) based on its spectral representation. By adopting the Breit-Wigner form for the spectral function rho(omega), we perform the best-fit analysis as a straightforward extension to the standard pole-mass analysis. The result indicates a significant broadening of the peak as Gamma(T) \sim 300 MeV as well as rather modest reduction of the peak center of about 100 MeV near T_c for the lowest 0^{++} glueball. The temporal correlators of the color-singlet modes corresponding to these glueballs above T_c are also investigated.Comment: This is the revised version using more gauge configurations near T_c. 25 pages, Latex2e, 22 figure