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P. Vranas, IBM Watson Research Lab 1 Gap domain wall fermions P. Vranas IBM Watson Research Lab

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P. Vranas, IBM Watson Research Lab 2 Memories :

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P. Vranas, IBM Watson Research Lab 3 Chiral symmetry restoration How much is the exponential rate of L s for chiral symmetry restoration? How does it depend on the lattice spacing (coupling)? First application of DWF was on the Schwinger model: P. Vranas, PRD57 (1998) 1415

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P. Vranas, IBM Watson Research Lab 4 At strong coupling R. Edwards, U. Heller, R. Narayanan, NPB 535 (1998) 403. Quenched QCD a -1 = 1 GeV

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P. Vranas, IBM Watson Research Lab 5 Zeroes of H(m 0 ) and instantons Instantons > a Instantons a Lattice dislocations Larger Brillouin zones Inactive * Quenched QCD a -1 = 2 GeV R. Edwards, U. Heller, R. Narayanan, NPB 535 (1998) 403.

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P. Vranas, IBM Watson Research Lab 6 Being stubborn :

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P. Vranas, IBM Watson Research Lab 7 Gap Domain Wall Fermions Improve DWF in the region 1 GeV < a -1 < 2 GeV. Since the problem occurs when H(m 0 ) is small multiply the Botzman weight with: det[H(m 0 )] = det[D(-m 0 )] This is the same as inserting Wilson fermions with heavy mass in the supercritical region (for example m 0 = 1.9). I will use 2 flavors. This will forbid zero crossings at m 0 and therefore enlarge the gap and reduce the residual mass. It will suppress instantons with size near the lattice spacing which are a lattice artifact (dislocations). Must check that the added Wilson fermions: - have hadron spectrum above the cutoff and are therefore irrelevant. - do not break parity (Aoki phase). - allow crossings due to instantons/anti-instantons with sizes > a (active topology).

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P. Vranas, IBM Watson Research Lab 8 P.M. Vranas, NATO Workshop, (2000) 11, Dubna, Russia, hep-lat/ P.M. Vranas, hep-lat/ T. Izubuchi, C. Dawson, Nucl. Phys. B (Proc. Suppl.) {\bf 106} (2002) 748. H. Fukaya, Ph.D. Thesis, Kyoto University, 2006, hep-lat/ H. Fukaya et. al. hep-lat/ H. Fukaya, S. Hashimoto,T. Kaneko, N. Yamada: Latt06, Chiral Symmetry 3.

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P. Vranas, IBM Watson Research Lab 9 Scale In order to compare DWF with GDWF I do quenched simulations at three lattice spacings: a -1= 1 GeV, 1.4 GeV and 2 GeV.

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P. Vranas, IBM Watson Research Lab 10 Quenched DWF, GDWF scale matching DWF data (diamonds) are from [RBC, PRD 69 (2004) ]. Matching is better than 5%. Use the rho to set the scale.

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P. Vranas, IBM Watson Research Lab 11 DWF GDWF a -1 = 1 GeV a -1 = 1.4 GeV a -1 = 2 GeV

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P. Vranas, IBM Watson Research Lab 12 Eigenvalue distribution Distribution of the 100 smallest eigenvalues from 110 independent configurations Here a -1 = 1.4 GeV DWF GDWF

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P. Vranas, IBM Watson Research Lab 13 The heavy Wilson flavors The pion (diamonds), rho (squares) and nucleon (stars) masses for 2 flavor dynamical Wilson flavors with mass = The straight line marks the cutoff

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P. Vranas, IBM Watson Research Lab 14 No parity breaking For two dynamical Wilson fermions with mass = They are outside the Aoki phase.

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P. Vranas, IBM Watson Research Lab 15 The residual mass m f = 0.02 and m 0 = 1.9 DWF GDWF DWF a -1 = 1.0 GeV a -1 = 1.4 GeV a -1 = 2.0 GeV

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P. Vranas, IBM Watson Research Lab 16 The residual mass dependence on m f L s = 16 m 0 = 1.9 GDWF

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P. Vranas, IBM Watson Research Lab 17 About that pion L s = 16 m 0 = 1.9 GDWF a -1 = 1.4 GeV a -1 = 1.0 GeV a -1 = 2.0 GeV

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P. Vranas, IBM Watson Research Lab 18 Pions in the rough seas I “imitate” a dynamical simulation with: L s = 24, m f = 0.005, V = 16 x 32, m 0 = 1.9, beta=4.6. I Find: a -1 = 1.356(75) GeV evaluated at m f. m res = (4) which is about 10% of m f. Finally: m pion = 140(40) MeV at m f =

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P. Vranas, IBM Watson Research Lab 19 Algorithmic and computational costs Simple to implement as an extension of Wilson and DWF: Add the two force terms and the two HMC Hamiltonians. For a 2 flavor DWF simulation it is an additional cost of 2 Dirac operators and therefore an additional 1/L s cost. For L s = 24 this is about 5%.

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P. Vranas, IBM Watson Research Lab 20 Net topology change GDWF may reduce the net-topology sampling of the traditional HMC because it forbids smooth deformations of topological objects. The eigenvalue flow can not cross m 0. This is only an algorithmic issue. We need algorithms that can tunnel between topological sectors. The same problem occurs in QCD anyway with or without GDWF. The QCD topological sectors are separated by energy barriers that become infinitely high as we approach the continuum. We have not been to small enough coupling yet in QCD to see the phenomenon. GDWF resemble continuous QCD in this way even more. In many cases net-topology change is not important provided one uses a large enough volume (see H. Leutwyler, A. Smilga, Phys. Rev D 46 (1992) 5607). It is important to see crossings in the larger-instanton regime since they confirm a topologically active vacuum. The net index may be fixed but the appearance/disappearance of instantons/anti-instantons is a property of the QCD vacuum and has to be there. Obviously then for large enough volumes cluster decomposition ensures correct physics.

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P. Vranas, IBM Watson Research Lab 21 An example from the Schwinger model P. Vranas PRD D57 (1998) 1415, hep-lat/

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P. Vranas, IBM Watson Research Lab 22 Net topology is practically fixed

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P. Vranas, IBM Watson Research Lab 23 GDWF: V=16 3 x 32, b=4.6, a -1 = 1.4 GeV

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P. Vranas, IBM Watson Research Lab 24 Curious ponderings An example of physics above the cutoff affecting the physics far below the cutoff. Do these “spectator” fermions point to something? The log of the 2 flavor Wilson determinant is an irrelevant operator and therefore a valid addition to the pure gauge action. Thinking about it this way Gap Fermions (GF) can be applied to all related methods (overlap etc..) and improvements. GDWF can be thought of as an extension of DWF by including the two additional Dirac operators along the fifth dimension diagonal behind the walls. This seems to be a natural extension of DWF with fermions beyond the walls that do not communicate directly with the fermions inside the walls. Their presence is felt only through their coupling to the gauge field in the bulk.

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P. Vranas, IBM Watson Research Lab 25 Open issues H is the transfer matrix Hamiltonian for DWF with continuous 5 th dimension. Here I used DWF with discrete 5 th dimension. The two Hamiltonians have the same zeroes so this is not an issue. However, H may be more effective for overlap. Since the DWF Hamiltonian is exactly known one may want to use “augmented” heavy Wilson flavors to see if they provide even more improvement. Although I used fairly large lattice spacings for the quenched study one must confirm with dynamical GDWF simulations. Why only 2 heavy Wilson flavors? Why not more? What happens then? GDWF may reduce the net-topology sampling of the traditional HMC. One can try instead of D 2 the operator D 2 + h 2 where h 2 is a real number.

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