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Non-perturbative improvement of nHYP smeared fermions R. Hoffmann In collaboration with A. Hasenfratz and S. Schaefer University of Colorado Peak Peak Mt. Wilson Wilson Mesa Wilson

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 2 Our action/setup Plaquette gauge action Wilson clover fermions coupled to nHYP smeared gauge field [→ talk by S. Schaefer] Schrödinger Functional setup: QCD with Dirichlet boundary conditions in time direction [ALPHA collaboration 1992-…] Provides IR cutoff → in principle m q = 0 is possible Useful for formulating non-perturbative improvement and renormalization conditions

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 3 “Smearing helps”… Want to quantify this Use plaquette gauge action → direct comparison to thin-link results Preparation for Schrödinger Functional runs with nHYP smearing We determine the clover coefficient non - perturbatively 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 [Lüscher et al. 1996] All thin link data we quote in the following is from Why?

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 4 nHYP smearing in the SF Need prescription to combine the HYP construction with SF boundary conditions Preserve boundary conditions through all levels in the construction: thin link nHYP link Intermediate links thin link boundary field SF boundary

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 5 Remark: smearing the tree-level BG field Any Abelian homogeneous boundary field induces a tree level background field Thus always link+staples = Projection/Normalization reduces this to Any projected/normalized smearing leaves the tree-level SF background field invariant Any projected/normalized smearing leaves the tree-level SF background field invariant

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 6 Determining the clover coefficient Follow the strategy [Lüscher et al. 1996] Use 2 PCAC quark masses in the SF w/ non-trivial BG field that are improved w/o knowing c A Simulate at several values of c SW, find point where masses “agree” All simulations at zero quark mass and Example: tree-level cutoff effect Sensitivity similar to thin link case

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 7 Results for c SW N.P. c sw remains small (even for coarse lattices) 1-loop nHYP seems to be tiny (0.266 for thin links)

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 8 What about c A ? Several methods available, see e.g. [Dürr & Della Morte 2003] quenched strategy (2 values) quenched strategy (2 values) slope method slope method gap method (wave functions) gap method (wave functions) Already with thin links: cutoff effects small after c SW is fixed e.g. method with nHYP: with increasing lattice spacing, cutoff effect actually decreases from its tree level value… Typical nHYP results at : No good signal with nHYP smearing after fixing c SW Might require larger volume/wave functions… But: How good are we already with c A = 0 ?

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 9 Test I :

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 10 Test II : 3 MeV band 5 MeV band

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 11 Test III : and thin nHYP Isovector vector current renormalization:

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 12 Exceptional configurations? Quenched SF w/ thin link clover requires to safely simulate … , no BG field: no BG field: Same at requires 25 MeV quark mass, but… : nHYP

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Non-perturbative improvement of nHYP smeared Wilson fermions R. Hoffmann, lat07 13 Conclusions / Outlook Successfully used nHYP smeared clover fermions in the Schrödinger functional framework (quenched for now) As expected, the non-perturbative c SW remains close to 1 with nHYP smeared clover fermions Finite renormalization constants seem to be ~1 The problem with exceptional configurations occurs only at much coarser lattice spacing Will be useful for non-perturbative improvement and renormalization with dynamical fermions N f > 0 is on the way…

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