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

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

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?

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

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

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

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)

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 ?

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

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

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

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

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…