THE TAMING OF

THE ARTIFACTSM I C H E L E P E P E

I N F N Sez. M i l a n o – B i c o c c a

Milan (I t a l y)

FERENC NIEDERMAYER

University of Bern

Bern (Switzerland)

UWE-JENS WIESE

University of Bern

Bern (Switzerland)

IN COLLABORATION WITH

P L A N O F T H E T A L K

Introduction

The problem of the lattice artifacts

Conclusions and outlook

A new approach tested for the O(3) spin model

Numerical results: further evidences

I n t r o d u c t i o n

Topological lattice actions: invariant against small deformations of the fields

Do not have the correct classical

continuum limit

Perturbation theory is not defined

Angle constrained action for O(N) spin models Seiler and Patrascioiu

Hasenbusch

W. Bietenholz, U. Gerber,

M. Pepe, and U.-J. Wiese

JHEP 1012:020, 2010

if

else

The action is pure entropy: no classical continuum limit but yields the correct quantum continuum limit as or

fixed

mass-gap in a system of size L step scaling functionM. Luscher , P. Weisz, U. Wolff (1991)

J. Balog, A. Hegedus (2004)

The problem of the lattice artifactsUniversality: any lattice action with the correct symmetries in the continuum works

BUTsome lattice actions give better approximations for continuum results for coarser

latticesNeed for reducing the effect of the lattice artifacts:

numerical effort in lattice QCD goes like L6 (quenched) or L10 (unquenched)

IDEA: modify the lattice operators by adding new terms, irrelevant in the continuum limit, that reduce the corrections due to finite lattice spacings

Minimizing discretization errors is a compromise between simplicity of the lattice

operators (numerically cheaper) and reduction of the lattice artifacts (smaller corrections)

Symanzik improvement: add systematically all local operators with the right symmetries of progressively larger dimension

local operators

K. Symanzik (1982-83)

Perfect action: method to remove any power-like dependence on the lattice spacing at the tree level for spectral quantities. It is based on Renormalization Group transformations.

P. Hasenfratz and F. Niedermayer (1994)

A new approach to remove the artifactsConstrained action: the standard n.n. action with the constraint on the maximal angle

if

elseA. Patrascioiu and E. Seiler (1992)

Choose L and fix a value for the constraint C

Fine tune β so that u0 attains the value 1.0595

At the same β, measure m(2L) 2L

If m(2L) 2L ≠1.26121035 then change C and back from start otherwise OK:

C is fixed and no longer changed

C = -0.345

Keeping fixed the estimated value for the constraint C we test the approach

For the O(3) model at θ≠0

J. Balog, private communication (2011)

O(4) O(8)

u0=1.0 u0=1.0595

PT or not PT?

The behavior predicted by Perturbation Theory for the standard action is in good agreement with the data.

J. Balog, F. Niedermayer,

and P. Weisz (2010)

depends only on the observable and not on the lattice action

depends on the lattice action and is computed in PT

The standard action and the constrained action are the same from the viewpoint of

Perturbation Theory

? Perturbation Theory is valid only at very very small lattice spacing: strongly non-perturbative regime

surprise!? P. Hasenfratz

Lattice2001 - Plenary

Conclusions

Outlook: the formulation of the method for non-Abelian gauge theories is straight-forward and it is the next, wishful, step we plan to take

The range of applicability of the approach is still under investigation

A new method to tame the nasty effects of the lattice artifacts is proposed and investigated

The method is fully non-perturbative and can be implemented very easily

Numerical results for the O(2), O(3) (θ=0, π/2, π), O(4) and O(8) spin models

show that the lattice artifacts can be efficiently removed below the 0.10/00 accuracy