non-perturbative improvement of nhyp smeared fermions r. hoffmann in collaboration with a....

Non-perturbative improvement of Non-perturbative improvement of nHYP smeared fermionsnHYP smeared fermions

R. HoffmannR. Hoffmann

In collaboration withIn collaboration with

A. Hasenfratz A. Hasenfratz andand S. Schaefer S. Schaefer

University of ColoradoUniversity of Colorado

PeakPeak

Mt. WilsonMt. WilsonWilson MesaWilson Mesa

WilsonWilson

Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07

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Our action/setupOur action/setup

Plaquette gauge actionPlaquette gauge action

Wilson clover fermions coupled toWilson clover fermions coupled tonHYPnHYP smeared gauge field smeared gauge field [[→→ talk by S. Schaefer] talk by S. Schaefer]

SchrSchröödinger Functionaldinger Functional setup: setup:QCD with Dirichlet boundary conditions in time directionQCD with Dirichlet boundary conditions in time direction

[[ALPHA collaboration ALPHA collaboration 1992-…1992-…]] Provides Provides IRIR cutoff → in principle cutoff → in principle mmq q = 0= 0 is possible is possible

Useful for formulating non-perturbative Useful for formulating non-perturbative improvementimprovement and andrenormalizationrenormalization conditions conditions


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““Smearing helps”… Want to quantify thisSmearing helps”… Want to quantify this

Use plaquette gauge action Use plaquette gauge action →→direct comparison to thin-link resultsdirect comparison to thin-link results

Preparation for SchrPreparation for Schrödinger Functionalödinger Functionalruns with runs with nHYPnHYP smearing smearing

We determine the clover coefficient nonWe determine the clover coefficient non -- perturbatively perturbatively using SF techniques in the quenched approximation…using SF techniques in the quenched approximation…

• Value of the clover coefficient

• Fluctuations of low Dirac modes (due to dislocations) → → appearance of exceptionals

• cutoff effects

[L[Lüscherüscher et al. 1996] et al. 1996]All thin link data we quote in the following is from All thin link data we quote in the following is from

Why?Why?


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nHYP smearing in the SFnHYP smearing in the SF

Need prescription to combine the HYP constructionNeed prescription to combine the HYP constructionwith SF boundary conditionswith SF boundary conditions

Preserve boundary conditions through all levels in the Preserve boundary conditions through all levels in the construction:construction:

thin linkthin linknHYPnHYP link link IntermediateIntermediatelinkslinks

thin link boundary fieldSF boundary


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Remark: smearing the tree-level BG fieldRemark: smearing the tree-level BG field

Any Any Abelian homogeneousAbelian homogeneous boundary field induces a boundary field induces atree level background fieldtree level background field

Thus Thus alwaysalways link+staples = link+staples =

Projection/Normalization reduces this to Projection/Normalization reduces this to

Any projected/normalized smearing leaves theAny projected/normalized smearing leaves thetree-level SF background field invarianttree-level SF background field invariant


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Determining the clover coefficientDetermining the clover coefficient Follow the Follow the strategy strategy [L[Lüscherüscher et al. 1996] et al. 1996]

Use 2 PCAC quark masses in the SF w/ non-trivial BG fieldUse 2 PCAC quark masses in the SF w/ non-trivial BG fieldthat are improved w/o knowing that are improved w/o knowing ccAA

Simulate at several values of Simulate at several values of ccSWSW, find point where masses “agree”, find point where masses “agree”

All simulations at zero quark mass andAll simulations at zero quark mass and

Example: Example:

tree-level cutoff effect

SensitivitySensitivitysimilar to similar to thin linkthin link case case

thin @ thin @


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Results for Results for ccSWSW

N.P. N.P. ccswsw remains smallremains small

(even for coarse lattices)(even for coarse lattices)

1-loop 1-loop nHYPnHYP seems to be tiny seems to be tiny

(0.266 for (0.266 for thin linksthin links))


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What about What about ccAA??

Several methods available, Several methods available, see e.g. see e.g. [D[Dürr & Della Morteürr & Della Morte 2003] 2003]

quenched strategy (2 quenched strategy (2 values) values)

slope methodslope method

gap method (wave functions)gap method (wave functions)

Already with Already with thin linksthin links: : cutoff effects small after cutoff effects small after ccSWSW is fixed is fixed

e.g. e.g. method with method with nHYPnHYP: : with increasing lattice spacing, cutoff effect actually with increasing lattice spacing, cutoff effect actually

decreasesdecreases from its tree level value… from its tree level value…

Typical Typical nHYPnHYP results at : results at : No good signal with No good signal with nHYPnHYP smearing after fixing smearing after fixing ccSWSW

Might require larger volume/wave functions…Might require larger volume/wave functions…

But: How good are we already with But: How good are we already with ccA A = 0 ?= 0 ?


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Test Test II: :


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Test Test IIII: :

3 MeV band3 MeV band5 MeV band5 MeV band


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Test Test IIIIII: and : and

thin

nHYPIsovector vector current renormalization:Isovector vector current renormalization:


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Exceptional configurations?Exceptional configurations?

Quenched SF w/ Quenched SF w/ thin linkthin link clover requires clover requires

to safely simulate to safely simulate ……

,,

no BG field:no BG field:

Same at requires 25 MeV quark mass, Same at requires 25 MeV quark mass, but…but…

::

nHYPnHYP


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Conclusions / OutlookConclusions / Outlook Successfully used Successfully used nHYPnHYP smeared clover fermions in the smeared clover fermions in the

SchrSchröödinger functional framework (quenched for now)dinger functional framework (quenched for now)

As expected, the As expected, the non-perturbative non-perturbative ccSWSW remains close remains close

to 1 with to 1 with nHYPnHYP smeared clover fermions smeared clover fermions

Finite renormalization constantsFinite renormalization constants seem to be ~1 seem to be ~1

The problem with The problem with exceptional configurationsexceptional configurationsoccurs only at occurs only at muchmuch coarser lattice spacing coarser lattice spacing

Will be useful for non-perturbative Will be useful for non-perturbative improvementimprovement and and renormalizationrenormalization with dynamical fermions with dynamical fermions

NNf f > 0 is on the way…> 0 is on the way…

non-perturbative improvement of nhyp smeared fermions r. hoffmann in collaboration with a....

Documents

nhyp smeared gauge field

loop nhyp

sf w nontrivial bg field

ir cutoff

projectednormalized

dirichlet boundary conditions

cutoff effects small

clover coefficientfollow