Comparing improved actions for SU(2)

Abstract

In order to help the user in choosing the right action a performance comparison is done for seven improved actions. Six of them are Symanzik improved, one at tree-level and two at one-loop, all with or without tadpole improvement. The seventh is an approximate fixed point action. Observables are static on-and off-axis two-body potentials and four-body binding energies, whose precision is compared when the same amount of computer time is used by the programs.We were motivated to consider using improved actions after noting the slowness of a four-quark flux distribution measurement code. In this case the lattice spacing has to be small, a ≈ 0.1 fm, to achieve sufficient resolution. In this work we compare actions at that scale and at a ≈ 0.2 fm. The actionsThe perturbative Symanzik approach to improvement is in this work represented by three actions; a tree-level version with a plaquette and a 1×2 rectangle[1] (abbreviation: S) and two one-loop actions, one with a 1 3 parallelogram (x,y,z,-x,-y,-z) [1,2] (S1) and the other with both parallelogram and a 2 × 2 large square [3] (S1S) as additional operators. Of these also the tadpole improved (TI) versions (STI, S1TI, S1STI) are considered. TI for the S1 action follows Ref.[4] using results for SU(2) in Ref.[2]. The non-perturbative approach to improvement is represented by a truncated fixed point action (FP) which includes first to fourth powers of the plaquette and the parallelogram [5]. The taskThe measurements consist of static two-quark potentials for R = 1, . . . , 6 on-axis, R = (1, 1), (2, 1), . . . , (3, 3) off-axis and the binding energies of four quarks at the corners of a regular * Presented by P. Pennanen, Petrus@hip.fi † Janne.Peisa@helsinki.fi tetrahedron, the cube surrounding it having sides of length R = 1, 2, 3. Here binding energies mean E 4 − 2V 2 , where E 4 is the energy of four quarks and 2V 2 the energy of the lowest-lying two-body pairing -see [6,7]. In order to separate the ground state a variational basis of fuzzing levels 13 and 2 is used. An update step consisted of four overrelaxations and one heatbath sweep, except for the FP case for which the latter was replaced by ten Metropolis sweeps. Table 1 shows the β values and corresponding scales used for the comparison. Scales were set by fitting plateau two-body potentials at R = 2, . . . , 6 with the continuum parameterization V fit = −e/R + b S R + V 0 and using Sommer's equation r 2 0 F (r 0 ) = c with c = 2.44, corresponding to r 0 ≈ 0.66 fm. These scales agree with the determination using √ b S = 440 MeV. The plateau was taken to be reached when the difference of potentials at T +1 and T was smaller than the bootstrap error on this difference.

QCD forces and heavy quark bound states

“…The static QCD potential has been determined to high accuracy in quenched lattice studies with Wilson [229,230,231,232,233,234,235,215,236,237,238,146,239,240,147,241,173,166,242,168] as well as various improved lattice actions [243,121,113,118,114] in SU(2) and SU(3) gauge theories. Results for QCD with sea quarks have been obtained in Refs.…”

“…However, in the quantum field theory case of interest, the coefficients are subject to radiative corrections, and have to be determined non-perturbatively to fully eliminate the O(a 2 ) lattice artefacts of the Wilson action. Although this has not been achieved yet, impressive results on static potentials [112,113,114], the glueball spectrum [115,116,117] and thermodynamics [118] have recently been obtained with Symanzik improved gluonic actions with coefficients, approximated by a mean field ("tadpole") estimate [119,120]. An alternative improved gluonic action that has been used in recent lattice studies [121] is the renormalisation group improved Iwasaki action [122,123].…”

Tadpole-improved SU(2) lattice gauge theory

“…(Simulations of the tadpole-improved SU(2) gauge theory on isotropic lattices have also recently been reported in Refs. [22,23]. )…”