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Register a new userDrastic Reduction of Cutoff Effects in 2-d Lattice O(N) Models
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We investigate the cutoff effects in 2-d lattice O(N) models for a variety of lattice actions, and we identify a class of very simple actions for which the lattice artifacts are extremely small. One action agrees with the standard action, except that it constrains neighboring spins to a maximal relative angle delta. We fix delta by demanding that a particular value of the step scaling function agrees with its continuum result already on a rather coarse lattice. Remarkably, the cutoff effects of the entire step scaling function are then reduced to the per mille level. This also applies to the theta-vacuum effects of the step scaling function in the 2-d O(3) model. The cutoff effects of other physical observables including the renormalized coupling and the mass in the isotensor channel are also reduced drastically. Another choice, the mixed action, which combines the standard quadratic with an appropriately tuned large quartic term, also has extremely small cutoff effects. The size of cutoff effects is also investigated analytically in 1-d and at N = infinity in 2-d.
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A variety of lattice discretisations of continuum actions has been considered, usually requiring the correct classical continuum limit. Here we discuss weird lattice formulations without that property, namely lattice actions that are invariant under most continuous deformations of the field configuration, in one version even without any coupling constants. It turns out that universality is powerful enough to still provide the correct quantum continuum limit, despite the absence of a classical limit, or a perturbative expansion. We demonstrate this for a set of O(N) models (or non-linear $sigma$-models). Amazingly, such weird lattice actions are not only in the right universality class, but some of them even have practical benefits, in particular an excellent scaling behaviour.
We present a lattice QCD study for the cutoff effects on nuclear forces. Two-nucleon forces are determined from Nambu-Bethe-Salpeter (NBS) wave functions using the HAL QCD method. Lattice QCD simulations are performed employing N_f = 2 clover fermion configurations at three lattice spacings of a = 0.108, 0.156, 0.215 fm on a fixed physical volume of L^3 x T = (2.5 fm)^3 x 5 fm with a large quark mass corresponding to m_pi = 1.1 GeV. We observe that while the discretization artifact appears at the short range part of potentials, it is suppressed at the long distance region. The cutoff dependence of the phase shifts and scattering length is also presented.
Discretization effects of lattice QCD are described by Symanziks effective theory when the lattice spacing, $a$, is small. Asymptotic freedom predicts that the leading asymptotic behavior is $sim a^n [bar g^2(a^{-1})]^{hatgamma_1} sim a^n left[frac{1}{-log(aLambda)}right]^{hatgamma_1}$. For spectral quantities, $n=d$ is given in terms of the (lowest) canonical dimension, $d+4$, of the operators in the local effective Lagrangian and $hatgamma_1$ is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix $gamma^{(0)}$. We determine $gamma^{(0)}$ for Yang-Mills theory ($n=2$) and discuss consequences in general and for perturbatively improved short distance observables. With the help of results from the literature, we also discuss the $n=1$ case of Wilson fermions with perturbative O$(a)$ improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to disappear faster than the naive $sim a^n$ and the log-corrections are a rather weak modification -- in contrast to the two-dimensional O(3) sigma model.
When analyzed in terms of the Symanzik expansion, lattice correlators of multi-local (gauge-invariant) operators with non-trivial continuum limit exhibit in maximally twisted lattice QCD ``infrared divergent cutoff effects of the type a^{2k}/(m_pi^2)^{h}, 2kgeq hgeq 1 (k,h integers), which tend to become numerically large when the pion mass gets small. We prove that, if the action is O(a) improved a` la Symanzik or, alternatively, the critical mass counter-term is chosen in some ``optimal way, these lattice artifacts are reduced to terms that are at worst of the order a^{2}(a^2/m_pi^2)^{k-1}, kgeq 1. This implies that the continuum extrapolation of lattice results is smooth at least down to values of the quark mass, m_q, satisfying the order of magnitude inequality m_q >a^2Lambda^3_{rm QCD}.
Supersymmetric lattice Ward-Takahashi identities are investigated perturbatively up to two-loop corrections for super doubler approach of $N=2$ lattice Wess-Zumino models in 1- and 2-dimensions. In this approach notorious chiral fermion doublers are treated as physical particles and momentum conservation is modified in such a way that lattice Leibniz rule is satisfied. The two major difficulties to keep exact lattice supersymmetry are overcome. This formulation defines, however, nonlocal field theory. Nevertheless we confirm that exact supersymmetry on the lattice is realized for all supercharges at the quantum level. Delicate issues of associativity are also discussed.
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