Continuum limit and improved action in lattice theories

Symanzik, K.

doi:10.1016/0550-3213(83)90468-6

Cited by 849 publications

(754 citation statements)

References 10 publications

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“…It is of the form Actually, this form of the local effective action for the O(n) nonlinear sigma model cannot be found in Symanzik's published papers [4,5]. But in the case of the φ 4 model an analogous local effective Lagrangian is introduced in [4] and in [5] an improved lattice action is presented for the O(n) sigma model. (3.5) is constructed using the φ 4 model analogy and has exactly the same terms as the improved action in [5].…”

Section: The Effective Lagrangianmentioning

confidence: 99%

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The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

2010

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Lattice artifacts in the 2d O(n) non-linear σ-model are expected to be of the form O(a 2 ), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter [1] we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a 2 ) artifacts which are especially large (ln 3 a) for n = 3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.

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Section: The Effective Lagrangianmentioning

confidence: 99%

“…In the early 80's Symanzik was working on the nature of lattice artifacts, in particular with respect to his improvement program [4,5,6,7,8]. In this paper the general theory is not discussed; we only consider in section 3 Symanzik's theory applied to the 2-dimensional O(n) σ-model 1 .…”

Section: Introductionmentioning

confidence: 99%

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

2010

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“…As the lattice spacing is small compared to the size of the proton and the inverse mass of the proton, these lattice artifacts are expected to provide a small modification to the N 2 LO terms we have determined. In addition, there are operators in the Symanzik action [80][81][82] that violate Lorentz symmetry as the calculations are performed on an underlying hypercubic grid. Such operators require the contraction of at least four Lorentz vectors in order to form a hypercubically-invariant, but Lorentz-violating, operator, for instance three derivatives and one electromagnetic field, or four derivatives.…”

Section: Lattice Artifactsmentioning

confidence: 99%

Finite-volume electromagnetic corrections to the masses of mesons, baryons, and nuclei

2014

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Now that Lattice QCD calculations are beginning to include QED, it is important to better understand how hadronic properties are modified by finite-volume QED effects. They are known to exhibit power-law scaling with volume, in contrast to the exponential behavior of finite-volume strong interaction effects. We use non-relativistic effective field theories describing the low-momentum behavior of hadrons to determine the finite-volume QED corrections to the masses of mesons, baryons and nuclei out to O 1/L 4 in a volume expansion, where L is the spatial extent of the cubic volume. This generalizes the previously determined expansion for mesons, and extends it by two orders in 1/L to include contributions from the charge radius, magnetic moment and polarizabilities of the hadron. We make an observation about direct calculations of the muon g − 2 in a finite volume.

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“…To carry it out, a theory based analytic form for the asymptotic dependence on the lattice spacing a is required. Usually a power behavior a and/or a 2 from Symanzik theory [6], based on all-order perturbation theory, is used. In [7,8] high precision simulations of the O(3)-model revealed however that the expected a 2 behavior does not prevail down to very small lattice spacings, and at present there seems to be no theoretical understanding for this.…”

Section: Introductionmentioning

confidence: 99%

Numerical confirmation of analytic predictions for the finite volume mass gap of the XY-model

et al. 2003

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Recent exact predictions for the massive scaling limit of the two dimensional XY-model are based on the equivalence with the sineGordon theory and include detailed results on the finite size behavior. The so-called step-scaling function of the mass gap is simulated with very high precision and found consistent with analytic results in the continuum limit. To come to this conclusion, an also predicted form of a logarithmic decay of lattice artifacts was essential to use for the extrapolation.

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Continuum limit and improved action in lattice theories

Cited by 849 publications

References 10 publications

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

Finite-volume electromagnetic corrections to the masses of mesons, baryons, and nuclei

Numerical confirmation of analytic predictions for the finite volume mass gap of the XY-model

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